Non-abelian  cohomology  and  Seifert–van Kampen  theorem

Contents
1.   Introduction 1   
2.   Group-valued cochains in dimensions $0$ and $1$ 6   
3.   Short  cochains 8   
4.   The standard  Seifert–van Kampen  theorem 11   
5.   Unions of  several  subsets 15
6.   van Kampen  theorems 17   
References   26

1. Introduction

The standard  Seifert–van Kampen  theorem.   The  Seifert–van Kampen  theorem  is  a common name for  theorems relating  the fundamental  group of  the union of  two or more spaces  to  the fundamental groups of these spaces and  their pairwise intersections,  under suitable  local  conditions.   The most  familiar  Seifert–van Kampen  theorem  deals with a space $X$ presented as  the union of  two open subsets $U\hskip 0.50003pt,\hskip 1.99997ptV$ such  that  $X\hskip 0.50003pt,\hskip 1.99997ptU\hskip 0.50003pt,\hskip 1.99997ptV$ and  the intersection  $U\hskip 1.00006pt\cap\hskip 1.00006ptV$ are path-connected and asserts  that  $\pi_{\hskip 0.70004pt1}\hskip 1.00006pt(\hskip 1.49994ptX\hskip 1.49994pt)$  is  equal  to  the free product  of  $\pi_{\hskip 0.70004pt1}\hskip 1.00006pt(\hskip 1.49994ptU\hskip 1.49994pt)$  and  $\pi_{\hskip 0.70004pt1}\hskip 1.00006pt(\hskip 1.49994ptV\hskip 1.49994pt)$  amalgamated over  $\pi_{\hskip 0.70004pt1}\hskip 1.00006pt(\hskip 1.49994ptU\hskip 1.00006pt\cap\hskip 1.00006ptV\hskip 1.49994pt)$.   The assumption  that  the subsets $U\hskip 0.50003pt,\hskip 1.99997ptV$ are open was,   probably,   first  introduced  by  R.  Crowell  and  R.  Fox  [C],  [CF].   Seifert  and  van Kampen  worked  with closed subsets,   and  Seifert  even with subcomplexes of  triangulated spaces.   In applications  the subsets are usually closed,  but  are deformation retracts of  their open neighborhoods  and can be replaced  by  such  neighborhoods.   At  the same  time  dealing  first  with open subsets,  or subsets with  the interiors covering $X$,  allows  to separate  the global  issues,   present  already for open subsets,   from  the  local  ones involved  in passing  from open subsets  to closed ones.  The present  paper deals with  the global  issues and  mostly with open subsets.

R.  Fox  related  [F]  that  he introduced  the name  “van Kampen  theorem”  for  the special  case of  path-connected  $U\hskip 0.50003pt,\hskip 1.99997ptV$ and  $U\hskip 1.00006pt\cap\hskip 1.00006ptV$,   overlooking  the fact  that  this case was proved  by  Seifert  [Se]  two years before  van Kampen’s  work  [vK].   Unfortunately,   this name  is  still  widely used  in  same sense.   This  led  to many claims  that  van Kampen  theorem  is  not  sufficient  to compute even  the fundamental  group of  the circle,  because  the circle cannot  be presented as  the union of  two path-connected subsets with path-connected  intersection.   In  fact,   the computation of  the fundamental  group of  the circle  is  a very special  case of  van Kampen  results.

van Kampen  theorems.   Let  $C\hskip 0.50003pt,\hskip 3.00003ptB$ be  topological  spaces such  that  $B$  is  path-connected and $C$ consists of  finite or countable number of  path-components.   Let $B_{\hskip 0.70004pt1}\hskip 1.00006pt,\hskip 1.99997ptB_{\hskip 0.70004pt2}\hskip 1.00006pt,\hskip 1.99997pt\ldots$ be a finite or countably  infinite collection of  closed subsets of  $C$ homeomorphic  to $B$ and such  that  every  path-connected component  of  $C$ contains at  least  one set  $B_{\hskip 0.70004pti}$.   Suppose  that  some homeomorphisms  $h_{\hskip 0.70004pti}\hskip 1.00006pt\colon\hskip 1.00006ptB_{\hskip 0.70004pti}\hskip 1.99997pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 1.99997ptB$  are fixed,   and  let  $A$  be  the quotient  space of  $C$  obtained  by  identifying  every  $B_{\hskip 0.70004pti}$  with  $B$  by  $h_{\hskip 0.70004pti}$.  van Kampen  [vK]  described  $\pi_{\hskip 0.70004pt1}\hskip 1.00006pt(\hskip 1.49994ptA\hskip 1.49994pt)$  in  terms of  $\pi_{\hskip 0.70004pt1}\hskip 1.00006pt(\hskip 1.49994ptC\hskip 1.49994pt)$  and  $\pi_{\hskip 0.70004pt1}\hskip 1.00006pt(\hskip 1.49994ptB\hskip 1.49994pt)$  and  of  homeomorphisms  $h_{\hskip 0.70004pti}$  under some  local  assumptions about  subspaces  $B_{\hskip 0.70004pti}$ and some  “niceness”  assumptions about  the  topological  spaces involved.   It  seems  that  the  latter are stronger  than necessary  and were borrowed  from  Lefschetz’s  treatment  of  the excision  property  of  homology  groups.   In  fact,   van Kampen  had no notion of  a quotient  space at  his disposal,   and  used a somewhat  cumbersome description of  relations between  $A$,  $C$,   and  $B_{\hskip 0.70004pti}$.   He started with $A$ and  $B$ and constructed $C$.   In any case,   if  $C\hskip 3.99994pt=\hskip 3.99994pt[\hskip 1.00006pt0\hskip 0.50003pt,\hskip 1.99997pt1\hskip 1.00006pt]$,  $B_{\hskip 0.70004pt1}\hskip 3.99994pt=\hskip 3.99994pt\{\hskip 1.49994pt0\hskip 1.49994pt\}$,  $B_{\hskip 0.70004pt2}\hskip 3.99994pt=\hskip 3.99994pt\{\hskip 1.49994pt1\hskip 1.49994pt\}$,  than $A$ is  a circle and $\pi_{\hskip 0.70004pt1}\hskip 1.00006pt(\hskip 1.49994ptA\hskip 1.49994pt)\hskip 3.99994pt=\hskip 3.99994pt\mathbb{Z}$ by  van Kampen  results.

In  the  last  section of  van Kampen’s  paper  “the path  is  shown  to a more general  theorem,   of  which however  the general  formulation  would  be more confusing  than helpful,   so  that  it  is  suppressed”.   van Kampen  singles out  two most  important  special  cases of  his  theorem,   both of  which are deal  with  the case of  two subsets $B_{\hskip 0.70004pt1}\hskip 1.00006pt,\hskip 1.99997ptB_{\hskip 0.70004pt2}$.   In  the first  one  the space $C$  is  assumed  to be path-connected.   In  the second special  case $C$  is  assumed  to consist  of  two path-connected components $C_{\hskip 0.70004pt1}\hskip 0.50003pt,\hskip 1.99997ptC_{\hskip 0.70004pt2}$ containing $B_{\hskip 0.70004pt1}\hskip 0.50003pt,\hskip 1.99997ptB_{\hskip 0.70004pt2}$ respectively.   See  [vK],   Corollaries  1  and  2.   The framework of  the second case  is  the same as  Seifert’s  one.   For a more detailed discussion of  van Kampen’s  results we refer  to  A.  Gramain  [G].

van Kampen’s  paper  [vK]  has  the reputation of  being  difficult.   It  is  indeed very densely written,   but  the present  author believes  that  the reason  is  different.   van Kampen  deals simultaneously with  three different  problems :   with  the  lack of  the notion of  quotient  spaces,   with  the  local  issues caused  by working with closed subsets  (for  the sake of  applications),   and,   finally,   with algebraic issues associated nowadays with  the  term  “van Kampen  theorem”.

van Kampen’s  framework and unions of  open subspaces.   In contrast  with  Seifert  and  the modern expositions,   van Kampen  worked not  with unions,   but  with some quotient  spaces.   There  is  a simple  trick allowing simultaneously  to pass from  van Kampen’s  framework  to unions of  open subspaces and  to separate  the  local  issues from  the global  ones.   Let

$$\quad B_{\hskip 0.70004pt\bullet}\hskip 3.99994pt=\hskip 3.99994pt\bigcup\nolimits_{\hskip 1.04996pti}\hskip 1.00006ptB_{\hskip 0.70004pti}$$

and  let  $h_{\hskip 0.70004pt\bullet}\hskip 1.00006pt\colon\hskip 1.00006ptB_{\hskip 0.70004pt\bullet}\hskip 1.99997pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 1.99997ptB$  be  the map defined  by  the maps  $h_{\hskip 0.70004pti}$.   Let  $X$  be  the cylinder of  the map  $h_{\hskip 0.70004pt\bullet}$.   In more details,  $X$  is  the result  of  glueing of  the subset

$$\quad C\hskip 1.99997pt\cup\hskip 1.99997ptB_{\hskip 0.70004pt\bullet}\hskip 1.00006pt\times\hskip 1.00006pt[\hskip 1.49994pt0\hskip 0.50003pt,\hskip 1.99997pt1\hskip 1.49994pt]\hskip 1.99997pt\subset\hskip 1.99997ptC\hskip 1.00006pt\times\hskip 1.00006pt[\hskip 1.49994pt0\hskip 0.50003pt,\hskip 1.99997pt1\hskip 1.49994pt]$$

to $B$  by  the map  $B_{\hskip 0.70004pt\bullet}\hskip 1.00006pt\times\hskip 1.00006pt1\hskip 1.00006pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 1.99997ptB$  induced  by $h_{\hskip 0.70004pt\bullet}$.   There  is  an obvious map  $X\hskip 1.99997pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 1.99997ptA$.   If  the pair  $(\hskip 1.49994ptC\hskip 0.50003pt,\hskip 3.00003ptB_{\hskip 0.70004pt\bullet}\hskip 1.49994pt)$  has  the homotopy  extension property ,   i.e.  the inclusion  $B_{\hskip 0.70004pt\bullet}\hskip 1.99997pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 1.99997ptC$  is  a cofibration,   then  this map  is  a  homotopy  equivalence.   Some weaker assumptions should  be sufficient  to ensure  that  $X\hskip 1.99997pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 1.99997ptA$  induces an isomorphism of  the fundamental  groups.   In order  to find  $\pi_{\hskip 0.70004pt1}\hskip 1.00006pt(\hskip 1.49994ptX\hskip 1.49994pt)$  one can  present  $X$  as  $X\hskip 3.99994pt=\hskip 3.99994ptU\hskip 1.00006pt\cup\hskip 1.00006ptV$,   where  $U\hskip 3.99994pt=\hskip 3.99994ptC\hskip 1.99997pt\cup\hskip 1.99997ptB_{\hskip 0.70004pt\bullet}\hskip 1.00006pt\times\hskip 1.00006pt[\hskip 1.49994pt0\hskip 0.50003pt,\hskip 1.99997pt2/3\hskip 1.49994pt)$ and  $V$  be  the image of  $B_{\hskip 0.70004pt\bullet}\hskip 1.00006pt\times\hskip 1.00006pt(\hskip 1.49994pt1/3\hskip 0.50003pt,\hskip 1.99997pt1\hskip 1.49994pt]$  in  $X$.   Then $U\hskip 1.00006pt\cap\hskip 1.00006ptV\hskip 3.99994pt=\hskip 3.99994ptB_{\hskip 0.70004pt\bullet}\hskip 1.00006pt\times\hskip 1.00006pt(\hskip 1.49994pt1/3\hskip 0.50003pt,\hskip 1.99997pt2/3\hskip 1.49994pt)$.   In  the second special  case of  van  Kampen  taking  as $U\hskip 0.50003pt,\hskip 1.99997ptV$  the images of  $C_{\hskip 0.70004pt1}\hskip 1.99997pt\cup\hskip 1.99997ptB_{\hskip 0.70004pt1}\hskip 1.00006pt\times\hskip 1.00006pt[\hskip 1.49994pt0\hskip 0.50003pt,\hskip 1.99997pt1\hskip 1.49994pt]\hskip 1.99997pt\cup\hskip 1.99997ptB_{\hskip 0.70004pt2}\hskip 1.00006pt\times\hskip 1.00006pt(\hskip 1.49994pt1/3\hskip 0.50003pt,\hskip 1.99997pt1\hskip 1.49994pt]$,  $C_{\hskip 0.70004pt2}\hskip 1.99997pt\cup\hskip 1.99997ptB_{\hskip 0.70004pt2}\hskip 1.00006pt\times\hskip 1.00006pt[\hskip 1.49994pt0\hskip 0.50003pt,\hskip 1.99997pt2/3\hskip 1.49994pt)$ respectively  results in path-connected  intersection  $U\hskip 1.00006pt\cap\hskip 1.00006ptV$.

Non-abelian cohomology  in  dimensions $0\hskip 0.50003pt,\hskip 1.00006pt1$.   P.  Olum  [O]  introduced  singular  cohomology  groups  $H^{\hskip 0.70004pt0}\hskip 1.00006pt(\hskip 1.49994ptX\hskip 0.50003pt,\hskip 1.99997ptA\hskip 1.00006pt;\hskip 1.99997pt\Pi\hskip 1.49994pt)$,  $H^{\hskip 0.70004pt1}\hskip 1.00006pt(\hskip 1.49994ptX\hskip 0.50003pt,\hskip 1.99997ptA\hskip 1.00006pt;\hskip 1.99997pt\Pi\hskip 1.49994pt)$  with possibly non-abelian  groups of  coefficients $\Pi$.   His  theory  is  modeled on  the  Eilenberg–Steenrod  axiomatic approach  to  the (co)homology  theory and  intended  for applications  to  homotopy classification of  mappings.   One of  his main  results was a  Mayer–Vietoris  sequence for such cohomology  groups.   As an application of  this  Mayer–Vietoris  sequence  Olum  presented a new proof  of  the second,   Seifert–like,   special  case of  van Kampen  theorem.   Adams  [A]  called  this proof  “simple and conceptual”.

P.  Olum  provided  neither a new proof  of  the first  special  case of  van Kampen  theorem,   nor even a new computation of  $\pi_{\hskip 0.70004pt1}\hskip 1.00006pt(\hskip 1.49994ptS^{\hskip 0.35002pt1}\hskip 1.49994pt)$.   As evidenced  by  R.  Crowell  [C]  and  R.  Fox  [F],   the computation of  $\pi_{\hskip 0.70004pt1}\hskip 1.00006pt(\hskip 1.49994ptS^{\hskip 0.35002pt1}\hskip 1.49994pt)$ wasn’t  considered  a worthwhile problem at  the  time,   in contrast  with  looking  for conceptual  proofs of  Seifert  and  van  Kampen  theorems.   Nevertheless,   seven  years  later  R.  Brown  opened  his paper  [$B\mathrm{r}_{\hskip 0.35002pt1}$]  with  the exclamation  “We present  another proof  that $\pi_{\hskip 0.70004pt1}\hskip 1.00006pt(\hskip 1.49994ptS^{\hskip 0.35002pt1}\hskip 1.49994pt)\hskip 3.99994pt=\hskip 3.99994pt\mathbb{Z}$!”.   Actually,   R.  Brown  [$B\mathrm{r}_{\hskip 0.35002pt1}$]  adapted  the method of  Olum  to  find  $\pi_{\hskip 0.70004pt1}\hskip 1.00006pt(\hskip 1.49994ptX\hskip 1.49994pt)$ when  $X\hskip 3.99994pt=\hskip 3.99994ptU\hskip 1.00006pt\cup\hskip 1.00006ptV$,   the interiors of  $U\hskip 0.50003pt,\hskip 1.99997ptV$  cover  $X$,   the subspaces $U\hskip 0.50003pt,\hskip 1.99997ptV$ are simply-connected,   and  the intersection  $U\hskip 1.00006pt\cap\hskip 1.00006ptV$  consists of  $n\hskip 1.99997pt+\hskip 1.99997pt1$  path-components.   Namely,   under  these assumptions $\pi_{\hskip 0.70004pt1}\hskip 1.00006pt(\hskip 1.49994ptX\hskip 1.49994pt)$  is  a free group on $n$ generators.   The proof  is  again simple and conceptual  and  includes an elegant  abstract  nonsense style argument  of  Adams.   We will  use  this argument  for  the same purpose.   See  Theorems  Non-abelian  cohomology  and  Seifert–van Kampen  theorem  and  Non-abelian  cohomology  and  Seifert–van Kampen  theorem.

R.  Brown  did not  develop  further these methods and embraced,   starting with  the paper  [$B\mathrm{r}_{\hskip 0.35002pt2}$]  and  the first  edition of  the book  [$B\mathrm{r}_{\hskip 0.35002pt3}$],   the ideology of  groupoids.   Later on  he wrote about  “all  the turgid stuff on nonabelian cohomology”.   See  [BHS],   Section  1.5.   In  the present  paper we  follow  “the road not  taken”  by  R.  Brown and  use  the non-abelian cohomology.

Crowell – Fox  version of  Seifert–van Kampen  theorem.   The results of  Seifert  [Se]  and  van Kampen  [vK]  were stated  in  terms of  generators and  relations.   In early  1950ies  Fox  reformulated  the standard  Seifert–van Kampen  theorem  (i.e.  the second special  case of  van Kampen)  in  terms of  direct  limits of  groups.   The proof  was worked out  by  R.  Crowell  [C]  and  led  to a more general  result  about  unions of  several  subsets.   Suppose  that  a  topological  space $X$  is  presented  as  the union  $X\hskip 3.99994pt=\hskip 3.99994pt\cup_{\hskip 0.70004pti}\hskip 1.00006ptU_{\hskip 0.35002pti}$  of  open  path-connected subsets $U_{\hskip 0.35002pti}\hskip 1.99997pt\subset\hskip 1.99997ptX$.   Suppose further  that  every subset  $U_{\hskip 0.35002pti}$ contains a fixed  base point $b\hskip 1.99997pt\in\hskip 1.99997ptX$,   and  that  the family of  subsets $U_{\hskip 0.35002pti}$  is  closed under  finite intersections.   Then  $\pi_{\hskip 0.70004pt1}\hskip 1.00006pt(\hskip 1.49994ptX\hskip 0.50003pt,\hskip 1.99997ptb\hskip 1.49994pt)$  is  a direct  limit  of  groups  $\pi_{\hskip 0.70004pt1}\hskip 1.00006pt(\hskip 1.49994ptU_{\hskip 0.35002pti}\hskip 0.50003pt,\hskip 1.99997ptb\hskip 1.49994pt)$  and  homomorphisms  induced  by  inclusions of  the form  $U_{\hskip 0.35002pti}\hskip 1.99997pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 1.99997ptU_{j}$.

An examination of  the proof  shows  that  that  it  is  sufficient  to assume  that  the family of  sets $U_{\hskip 0.35002pti}$  is  closed  under  taking  the intersection of  pairs of  sets and  that  intersections of  $\leqslant\hskip 1.99997pt4$  sets $U_{\hskip 0.35002pti}$  are path-connected.   R.  Brown  and  A.R.  Salleh  [BS],   working  in  the groupoid  language,   showed  in  1984  that  the  last  condition can  be relaxed.   Namely,   it  is  sufficient  to assume  that  intersections of  $\leqslant\hskip 1.99997pt3$  sets $U_{\hskip 0.35002pti}$  are path-connected.   The groupoid  language  is  irrelevant  for  this improvement.   The corresponding argument  is  based on  the fact  that  the  Lebesgue  covering dimension of  the disc  is  $\leqslant\hskip 1.99997pt2$.   A proof  written  in  the usual  language of  groups  is  contained  in  Hatcher’s  textbook  [H].   See  [H],   Theorem  1.20.   As we will  see in  the proof  of  Theorem  Non-abelian  cohomology  and  Seifert–van Kampen  theorem,   the  Lebesgue dimension of  the disc  is  also irrelevant.

The present  paper .   Our  first  goal  is  to present  a simple and elementary  proof  of  the standard  Seifert–van Kampen  theorem  based on  the ideas of  Olum  [O].   In contrast  with  Olum,   we do not  discuss analogues of  the  Eilenberg–Steenrod  axioms and  the  Mayer–Vietoris  sequence.   Instead,   we work  mostly  with non-abelian cochains an cocycles and use  the standard  tool  of  subdividing  the unit  square into small  squares.   The unpleasant  part  of  standard  proofs,   the need  to keep  track of  paths connecting  the subdivision points  to  a base point,   is  absorbed  by  the  notion of  cohomologous cocycles.   The resulting  proof  requires the same prerequisites as  the proof  in any  modern  textbook.   It  occupies  Sections  Non-abelian  cohomology  and  Seifert–van Kampen  theorem – Non-abelian  cohomology  and  Seifert–van Kampen  theorem.   The key  geometric result  is  Theorem  Non-abelian  cohomology  and  Seifert–van Kampen  theorem,   proved  by  subdividing  the unit  square.

In  Section  Non-abelian  cohomology  and  Seifert–van Kampen  theorem  we extend  the methods of  Section  Non-abelian  cohomology  and  Seifert–van Kampen  theorem  in order  to prove  the  Crowell – Fox  version of  Seifert–van Kampen  theorem and  its  Brown-Salleh  improvement.   The proof  shows  that  the reason behind  the condition  that  triple intersections are path-connected  is  purely combinatorial.   There  is  no need  to subdivide  the unit  square into small  rectangles without  fourfold  intersections.   This contrasts with  [BS],  and with  [H],   the proof  of  Theorem  1.20.

Finally,   in  Section  Non-abelian  cohomology  and  Seifert–van Kampen  theorem  we extend  the methods of  the previous sections  to  determine  $\pi_{\hskip 0.70004pt1}\hskip 1.00006pt(\hskip 1.49994ptX\hskip 1.49994pt)$  when $X\hskip 3.99994pt=\hskip 3.99994ptU\hskip 1.00006pt\cup\hskip 1.00006ptV$,   the subsets $U\hskip 0.50003pt,\hskip 1.99997ptV$  are open,   and  the intersection  $U\hskip 1.00006pt\cap\hskip 1.00006ptV$  consists of  finitely  many  path-components.   This proves  the open sets version of  the main  results of  van Kampen  (and,   in  particular ,   computes  the fundamental  group of  the circle).   We deal  with  the case when  $U\hskip 1.00006pt\cap\hskip 1.00006ptV$  consists of  two path-components  in  Theorem  Non-abelian  cohomology  and  Seifert–van Kampen  theorem,   and  with  the general  case in  Theorem  Non-abelian  cohomology  and  Seifert–van Kampen  theorem.   The same methods work in  the case of  infinitely many components.

Small  simplices and short  paths.   Let  $\mathcal{U}$  be an open covering of  a  topological  space $X$.   Let  us call  a singular simplex  in $X$  small  if  its image  is  contained  in some $U\hskip 1.99997pt\in\hskip 1.99997pt\mathcal{U}$.   The usual  definition of  the singular homology and cohomology groups with coefficients in an abelian group $\Pi$ can be modified  by considering only small  singular simplices.   We will  indicate  this modification by  the subscript  $\mathcal{U}$,  as  in  $H^{\hskip 0.35002ptn}_{\hskip 0.70004pt\mathcal{U}}\hskip 1.00006pt(\hskip 1.49994ptX\hskip 0.50003pt,\hskip 1.99997ptA\hskip 1.00006pt;\hskip 1.99997pt\Pi\hskip 1.49994pt)$.   There  is  an obvious map

(1)

$$\quad H^{\hskip 0.35002ptn}\hskip 1.00006pt(\hskip 1.49994ptX\hskip 0.50003pt,\hskip 1.99997ptA\hskip 1.00006pt;\hskip 1.99997pt\Pi\hskip 1.49994pt)\hskip 1.99997pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 1.99997ptH^{\hskip 0.35002ptn}_{\hskip 0.70004pt\mathcal{U}}\hskip 1.00006pt(\hskip 1.49994ptX\hskip 0.50003pt,\hskip 1.99997ptA\hskip 1.00006pt;\hskip 1.99997pt\Pi\hskip 1.49994pt)\hskip 1.99997pt,$$

and  by a fundamental  theorem,   essentially due  to  Eilenberg  [E],   this map  is  an  isomorphism.   See,   for example,   Proposition  2.21  in  Hatcher’s  textbook  [H].   This results  is  the key  geometric step  to  the basic results of  the singular (co)homology  theory such as  the excision property and  the  Mayer–Vietoris  sequence.   The geometric part  of  the proof  is  based on subdividing simplices into smaller ones,  the barycentric subdivision  being  the standard  tool.   The algebraic part  is  an elegant  construction of  chain homotopies due  to  Eilenberg  [E].

P.  Olum  observed  that  the same arguments work  for cohomology sets with non-abelian coefficients $\Pi$ in dimensions $0\hskip 0.50003pt,\hskip 1.99997pt1$.   In  fact,   the algebraic part  of  the proof  is  even simpler .   See  [O],   the proof  of  (2.5).   As in  the (co)homology  theory,   Olum  uses  the iterated  barycentric subdivisions of  simplices of  dimension $\leqslant\hskip 1.99997pt2$.   The present  author believes  that  this unification with  the (co)homology  theory  is  an  important  advantage of  the non-abelian cohomology approach  to  Seifert–van Kampen  theorems.

In  the context  of  fundamental  groups the barycentric subdivisions are not  quite natural,   at  least  if  the  theory of  fundamental  groups precedes  the (co)homology  theory,   as it  is  usually  the case.   By  this reason  we replaced  singular simplices by  paths and  homotopies.   We call  a path or a homotopy  short  if  its image  is  contained  in some $U\hskip 1.99997pt\in\hskip 1.99997pt\mathcal{U}$.   The iterated  barycentric subdivision of  triangles are replaced  by  the much simpler  tool  of  subdividing a square into smaller squares.   See  the proof  of  Theorem  Non-abelian  cohomology  and  Seifert–van Kampen  theorem,   the analogue of  the isomorphism  (1).

Notations and conventions.   The interval  $[\hskip 1.00006pt0\hskip 0.50003pt,\hskip 1.99997pt1\hskip 1.00006pt]$  is  often  denoted  by  $I$.   A  path  in a  topological  spase  $X$  is  a map  $I\hskip 1.99997pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 1.99997ptX$.   If  $p\hskip 0.50003pt,\hskip 1.99997ptq$ are  two paths and  $p\hskip 1.00006pt(\hskip 1.00006pt1\hskip 1.00006pt)\hskip 3.99994pt=\hskip 3.99994ptq\hskip 1.00006pt(\hskip 1.00006pt0\hskip 1.00006pt)$,   then  the product  $p\hskip 1.00006pt\cdot\hskip 1.00006ptq$  is  defined  in  the usual  manner  by  following first  $p$ and  then $q$.   If  $p$  is  a  path,   then  $\overline{p}$  is  defined  by  $\overline{p}\hskip 1.49994pt(\hskip 1.00006pts\hskip 1.49994pt)\hskip 3.99994pt=\hskip 3.99994ptp\hskip 1.00006pt(\hskip 1.00006pt1\hskip 1.99997pt-\hskip 1.99997pts\hskip 1.49994pt)$.   Homotopies of  paths  are assumed  to be  homotopies relatively  to  the boundary  $\{\hskip 1.49994pt0\hskip 0.50003pt,\hskip 1.99997pt1\hskip 1.49994pt\}$.   We write  $p\hskip 1.99997pt\sim\hskip 1.99997ptq$  when  paths  $p\hskip 0.50003pt,\hskip 3.00003ptq$  are homotopic.

A  reparametrization  of  a path  $p$ is  a  path of  the form  $p\hskip 1.00006pt\circ\hskip 1.00006pt\varphi$ where  $\varphi\hskip 1.00006pt\colon\hskip 1.00006ptI\hskip 1.99997pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 1.99997ptI$  is  a  map such  that $\varphi\hskip 1.00006pt(\hskip 1.00006pt0\hskip 1.00006pt)\hskip 3.99994pt=\hskip 3.99994pt0\hskip 0.50003pt,\hskip 3.99994pt\varphi\hskip 1.00006pt(\hskip 1.00006pt1\hskip 1.00006pt)\hskip 3.99994pt=\hskip 3.99994pt1$.   Clearly,  any  reparametrization of $p$  is  homotopic  to $p$.

Let  $p_{\hskip 0.70004pt1}\hskip 1.00006pt,\hskip 3.99994ptp_{\hskip 0.70004pt2}\hskip 1.00006pt,\hskip 3.99994pt\ldots\hskip 1.00006pt,\hskip 3.99994ptp_{\hskip 0.70004ptk}$  be paths  in  $X$  such  that  the products $p_{\hskip 0.70004pti}\hskip 1.00006pt\cdot\hskip 1.00006ptp_{\hskip 0.70004pti\hskip 0.70004pt+\hskip 0.70004pt1}$  are defined,   i.e.  such  that  $p_{\hskip 0.70004pti}\hskip 1.00006pt(\hskip 1.00006pt1\hskip 1.00006pt)\hskip 3.99994pt=\hskip 3.99994ptp_{\hskip 0.70004pti\hskip 0.70004pt+\hskip 0.70004pt1}\hskip 1.00006pt(\hskip 1.00006pt0\hskip 1.00006pt)$  for every  $i\hskip 1.99997pt\leqslant\hskip 1.99997ptk\hskip 1.99997pt-\hskip 1.99997pt1$.   Let  us define  $p\hskip 3.99994pt=\hskip 3.99994ptp_{\hskip 0.70004pt1}\hskip 1.00006pt\cdot\hskip 1.00006ptp_{\hskip 0.70004pt2}\hskip 1.00006pt\cdot\hskip 1.00006pt\ldots\hskip 1.00006pt\cdot\hskip 1.00006ptp_{\hskip 0.70004ptk}$  by  the rule

$$\quad p\hskip 1.00006pt(\hskip 1.00006pts\hskip 1.49994pt)\hskip 3.99994pt=\hskip 3.99994ptp_{\hskip 0.70004pti}\hskip 1.00006pt\bigl{(}\hskip 1.49994ptk\hskip 1.00006pts\hskip 1.99997pt-\hskip 1.99997pti\hskip 1.99997pt+\hskip 1.99997pt1\hskip 1.49994pt\bigr{)}\quad\ \mbox{for}\quad\ (\hskip 1.49994pti\hskip 1.99997pt-\hskip 1.99997pt1\hskip 1.49994pt)/k\hskip 1.99997pt\leqslant\hskip 1.99997pts\hskip 1.99997pt\leqslant\hskip 1.99997pti\hskip 0.50003pt/k\hskip 3.00003pt.$$

Clearly ,  $p_{\hskip 0.70004pt1}\hskip 1.00006pt\cdot\hskip 1.00006ptp_{\hskip 0.70004pt2}\hskip 1.00006pt\cdot\hskip 1.00006pt\ldots\hskip 1.00006pt\cdot\hskip 1.00006ptp_{\hskip 0.70004ptk}$  differs  by  a  reparametrization  from each  product  obtained  by  placing  parentheses  into  the expression  $p_{\hskip 0.70004pt1}\hskip 1.00006pt\cdot\hskip 1.00006ptp_{\hskip 0.70004pt2}\hskip 1.00006pt\cdot\hskip 1.00006pt\ldots\hskip 1.00006pt\cdot\hskip 1.00006ptp_{\hskip 0.70004ptk}$.   Therefore  the product  $p_{\hskip 0.70004pt1}\hskip 1.00006pt\cdot\hskip 1.00006ptp_{\hskip 0.70004pt2}\hskip 1.00006pt\cdot\hskip 1.00006pt\ldots\hskip 1.00006pt\cdot\hskip 1.00006ptp_{\hskip 0.70004ptk}$  is  homotopic  to every  such  product  and  may serve as a  partial  replacement  of  associativity .

2. Group-valued  cochains  in  dimensions  $0$  and  $1$

Cochains and cocycles.   Let  $X$  be a  path-connected  space,  $Y\hskip 1.99997pt\subset\hskip 1.99997ptX$,   and $b\hskip 1.99997pt\in\hskip 1.99997ptY$.   Let  us  fix a  group  $G$  and denote by $1$  the unit  of  $G$.   Let  us denote by $P\hskip 1.00006pt(\hskip 1.49994ptX\hskip 1.49994pt)$  the set  of  all  paths in  $X$,   i.e.  the set  of  all  continuous maps  $[\hskip 1.00006pt0\hskip 0.50003pt,\hskip 1.99997pt1\hskip 1.00006pt]\hskip 1.99997pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 1.99997ptX$.

A $0$-cochain  of  the pair  $(\hskip 1.49994ptX\hskip 0.50003pt,\hskip 3.00003ptY\hskip 1.49994pt)$  is  a map  $c\hskip 1.00006pt\colon\hskip 1.00006ptX\hskip 1.99997pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 1.99997ptG$  such  that  $c\hskip 1.49994pt(\hskip 1.00006pty\hskip 1.49994pt)\hskip 3.99994pt=\hskip 3.99994pt1$  if  $y\hskip 1.99997pt\in\hskip 1.99997ptY$,   and a $1$-cochain  of  $(\hskip 1.49994ptX\hskip 0.50003pt,\hskip 3.00003ptY\hskip 1.49994pt)$  is  a map  $u\hskip 1.00006pt\colon\hskip 1.00006ptP\hskip 1.00006pt(\hskip 1.49994ptX\hskip 1.49994pt)\hskip 1.99997pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 1.99997ptG$  such  that  $u\hskip 1.49994pt(\hskip 1.00006ptp\hskip 1.49994pt)\hskip 3.99994pt=\hskip 3.99994pt1$  if  $p\hskip 1.99997pt\in\hskip 1.99997ptP\hskip 1.00006pt(\hskip 1.49994ptY\hskip 1.49994pt)$.   The set  of  all  $n$-cochains of  $(\hskip 1.49994ptX\hskip 0.50003pt,\hskip 3.00003ptY\hskip 1.49994pt)$,   where $n\hskip 3.99994pt=\hskip 3.99994pt0$  or  $1$,   is  denoted  by  $C^{\hskip 0.70004ptn}\hskip 1.00006pt(\hskip 1.49994ptX\hskip 0.50003pt,\hskip 3.00003ptY\hskip 1.49994pt)$.   Of  course,  $C^{\hskip 0.70004ptn}\hskip 1.00006pt(\hskip 1.49994ptX\hskip 0.50003pt,\hskip 3.00003ptY\hskip 1.49994pt)$  depends also on  $G$.   A $1$-cochain $u$ is  called a cocycle  if  $u\hskip 1.49994pt(\hskip 1.00006ptp\hskip 1.49994pt)\hskip 3.99994pt=\hskip 3.99994ptu\hskip 1.49994pt(\hskip 1.00006ptq\hskip 1.49994pt)$  when  $p\hskip 3.99994pt\sim\hskip 3.99994ptq$  and $u\hskip 1.49994pt(\hskip 1.00006ptp\hskip 1.00006pt\cdot\hskip 1.00006ptq\hskip 1.49994pt)\hskip 3.99994pt=\hskip 3.99994ptu\hskip 1.49994pt(\hskip 1.00006ptp\hskip 1.49994pt)\hskip 1.00006pt\cdot\hskip 1.00006ptu\hskip 1.49994pt(\hskip 1.00006ptq\hskip 1.49994pt)$  when  the product  $p\hskip 1.00006pt\cdot\hskip 1.00006ptq$  is  defined.   The set  of  all  cocycles of  $(\hskip 1.49994ptX\hskip 0.50003pt,\hskip 3.00003ptY\hskip 1.49994pt)$  is  denoted  by  $Z^{\hskip 0.70004pt1}\hskip 1.00006pt(\hskip 1.49994ptX\hskip 0.50003pt,\hskip 3.00003ptY\hskip 1.49994pt)$.

The case of  $Y\hskip 3.99994pt=\hskip 3.99994pt\{\hskip 1.00006ptb\hskip 1.49994pt\}$  is  the most  important  one.   We will  abbreviate  $(\hskip 1.49994ptX\hskip 0.50003pt,\hskip 3.00003pt\{\hskip 1.00006ptb\hskip 1.49994pt\}\hskip 1.49994pt)$  as  $(\hskip 1.49994ptX\hskip 0.50003pt,\hskip 3.00003ptb\hskip 1.49994pt)$.   The case of  discrete subsets  $Y$  is  also important.   In  this case every  $p\hskip 1.99997pt\in\hskip 1.99997ptP\hskip 1.00006pt(\hskip 1.49994ptY\hskip 1.49994pt)$  is  a constant  path and  hence  $p\hskip 3.99994pt\sim\hskip 3.99994ptp\hskip 1.00006pt\cdot\hskip 1.00006ptp$.   Then  $u\hskip 1.49994pt(\hskip 1.00006ptp\hskip 1.49994pt)\hskip 3.99994pt=\hskip 3.99994pt1$  for every $1$-cocycle $u$ and  $p\hskip 1.99997pt\in\hskip 1.99997ptP\hskip 1.00006pt(\hskip 1.49994ptY\hskip 1.49994pt)$.

Cohomology .   The point-wise multiplication of  maps $X\hskip 1.00006pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 1.00006ptG$ turns $C^{\hskip 0.70004pt0}\hskip 1.00006pt(\hskip 1.49994ptX\hskip 0.50003pt,\hskip 1.99997ptY\hskip 1.49994pt)$ into a  group.   The group  $C^{\hskip 0.70004pt0}\hskip 1.00006pt(\hskip 1.49994ptX\hskip 0.50003pt,\hskip 3.00003ptY\hskip 1.49994pt)$ acts on $C^{\hskip 0.70004pt1}\hskip 1.00006pt(\hskip 1.49994ptX\hskip 0.50003pt,\hskip 3.00003ptY\hskip 1.49994pt)$  by  the rule

$$\quad(\hskip 1.49994ptc\hskip 1.00006pt\bullet\hskip 1.00006ptu\hskip 1.49994pt)\hskip 1.49994pt(\hskip 1.00006ptp\hskip 1.49994pt)\hskip 3.99994pt=\hskip 3.99994ptc\hskip 1.49994pt(\hskip 1.49994ptp\hskip 1.00006pt(\hskip 1.00006pt0\hskip 1.00006pt)\hskip 1.49994pt)\hskip 1.00006pt\cdot\hskip 1.00006ptu\hskip 1.49994pt(\hskip 1.00006ptp\hskip 1.49994pt)\hskip 1.00006pt\cdot\hskip 1.00006ptc\hskip 1.49994pt(\hskip 1.49994ptp\hskip 1.00006pt(\hskip 1.00006pt1\hskip 1.00006pt)\hskip 1.49994pt)^{\hskip 0.70004pt-\hskip 0.70004pt1}\hskip 3.00003pt,$$

where  $c\hskip 1.99997pt\in\hskip 1.99997ptC^{\hskip 0.70004pt0}\hskip 1.00006pt(\hskip 1.49994ptX\hskip 0.50003pt,\hskip 3.00003ptY\hskip 1.49994pt)$,  $u\hskip 1.99997pt\in\hskip 1.99997ptC^{\hskip 0.70004pt1}\hskip 1.00006pt(\hskip 1.49994ptX\hskip 0.50003pt,\hskip 3.00003ptY\hskip 1.49994pt)$,   and  $p\hskip 1.99997pt\in\hskip 1.99997ptP\hskip 1.00006pt(\hskip 1.49994ptX\hskip 1.49994pt)$.   An  immediate verification shows  that  $c\hskip 1.00006pt\bullet\hskip 1.00006pt(\hskip 1.49994ptd\hskip 1.00006pt\bullet\hskip 1.00006ptu\hskip 1.49994pt)\hskip 3.99994pt=\hskip 3.99994pt(\hskip 1.00006ptc\hskip 1.00006pt\cdot\hskip 1.00006ptd\hskip 1.49994pt)\hskip 1.00006pt\bullet\hskip 1.00006ptu$  and  hence  $(\hskip 1.00006ptc\hskip 0.50003pt,\hskip 3.00003ptu\hskip 1.49994pt)\hskip 1.99997pt\longmapsto\hskip 1.99997ptc\hskip 1.00006pt\bullet\hskip 1.00006ptu$  is  indeed an action.   Another  easy verification shows  that  $c\hskip 1.00006pt\bullet\hskip 1.00006ptu$  is  a cocycle  if  $u$  is  a cocycle.   Two cocycles  $z\hskip 0.50003pt,\hskip 3.00003ptu$  are called  cohomologous  if  they  belong  to  the same orbit  of  the action of  $C^{\hskip 0.70004pt0}\hskip 1.00006pt(\hskip 1.49994ptX\hskip 0.50003pt,\hskip 3.00003ptY\hskip 1.49994pt)$,   i.e.  if  $z\hskip 3.99994pt=\hskip 3.99994ptc\hskip 1.00006pt\bullet\hskip 1.00006ptu$  for some  $c\hskip 1.99997pt\in\hskip 1.99997ptC^{\hskip 0.70004pt0}\hskip 1.00006pt(\hskip 1.49994ptX\hskip 0.50003pt,\hskip 3.00003ptY\hskip 1.49994pt)$.  Being cohomologous  is  an equivalence relation on  the set  $Z^{\hskip 0.70004pt1}\hskip 1.00006pt(\hskip 1.49994ptX\hskip 0.50003pt,\hskip 3.00003ptY\hskip 1.49994pt)$.   The set  of  equivalence classes  is  denoted  by  $H^{\hskip 0.70004pt1}\hskip 1.00006pt(\hskip 1.49994ptX\hskip 0.50003pt,\hskip 3.00003ptY\hskip 1.49994pt)$.   The equivalence class of  a cocycle $u$  is  called  the  cohomology  class  of  $u$ and denoted  by $[\hskip 1.00006ptu\hskip 0.50003pt\hskip 1.00006pt]$.   If  $G$  is  not  abelian,  $H^{\hskip 0.70004pt1}\hskip 1.00006pt(\hskip 1.49994ptX\hskip 0.50003pt,\hskip 3.00003ptY\hskip 1.49994pt)$ has no  natural  group structure,  but  has a distinguished element,  called  trivial  cohomology  class,   namely ,   the equivalence class $1$ of  the cocycle $\mathbb{1}$ mapping every  path  to $1\hskip 1.99997pt\in\hskip 1.99997ptG$.

Cohomology  classes and  homomorphisms.   It  turns out  that  there  is  a canonical  bijection  between  $H^{\hskip 0.70004pt1}\hskip 1.00006pt(\hskip 1.49994ptX\hskip 0.50003pt,\hskip 3.00003ptb\hskip 1.00006pt)$  and  the set  $\operatorname{Hom}\hskip 1.49994pt(\hskip 1.49994pt\pi_{\hskip 0.70004pt1}\hskip 1.00006pt(\hskip 1.49994ptX\hskip 0.50003pt,\hskip 1.99997ptb\hskip 1.49994pt)\hskip 0.50003pt,\hskip 3.00003ptG\hskip 1.49994pt)$  of  homomorphisms  $\pi_{\hskip 0.70004pt1}\hskip 1.00006pt(\hskip 1.49994ptX\hskip 0.50003pt,\hskip 1.99997ptb\hskip 1.49994pt)\hskip 1.99997pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 1.99997ptG$.   Let  $L\hskip 1.49994pt(\hskip 1.49994ptX\hskip 0.50003pt,\hskip 1.99997ptb\hskip 1.49994pt)$  be  the set  of  loops in  $X$  based at  $b$,   and  let  us  interpret  homomorphisms  $\pi_{\hskip 0.70004pt1}\hskip 1.00006pt(\hskip 1.49994ptX\hskip 0.50003pt,\hskip 1.99997ptb\hskip 1.49994pt)\hskip 1.99997pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 1.99997ptG$  as maps  $h\hskip 1.00006pt\colon\hskip 1.00006ptL\hskip 1.49994pt(\hskip 1.49994ptX\hskip 0.50003pt,\hskip 1.99997ptb\hskip 1.49994pt)\hskip 1.99997pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 1.99997ptG$  such  that  $h\hskip 1.49994pt(\hskip 1.00006ptp\hskip 1.49994pt)\hskip 3.99994pt=\hskip 3.99994pth\hskip 1.49994pt(\hskip 1.00006ptq\hskip 1.49994pt)$  if  $p\hskip 3.99994pt\sim\hskip 3.99994ptq$,   and such  that  $h\hskip 1.49994pt(\hskip 1.00006ptp\hskip 1.00006pt\cdot\hskip 1.00006ptq\hskip 1.49994pt)\hskip 3.99994pt=\hskip 3.99994pth\hskip 1.49994pt(\hskip 1.00006ptp\hskip 1.49994pt)\hskip 1.00006pt\cdot\hskip 1.00006pth\hskip 1.49994pt(\hskip 1.00006ptq\hskip 1.49994pt)$  for every  $p\hskip 0.50003pt,\hskip 3.00003ptq\hskip 1.99997pt\in\hskip 3.00003ptL\hskip 1.49994pt(\hskip 1.49994ptX\hskip 0.50003pt,\hskip 1.99997ptb\hskip 1.49994pt)$.   Let  $u\hskip 1.99997pt\in\hskip 1.99997ptZ^{\hskip 0.70004pt1}\hskip 1.00006pt(\hskip 1.49994ptX\hskip 0.50003pt,\hskip 3.00003ptb\hskip 1.00006pt)$.   Since  $u$  is  a cocycle,   the restriction of  $u$  to  $L\hskip 1.49994pt(\hskip 1.49994ptX\hskip 0.50003pt,\hskip 1.99997ptb\hskip 1.49994pt)\hskip 1.99997pt\subset\hskip 1.99997ptP\hskip 1.00006pt(\hskip 1.49994ptX\hskip 1.49994pt)$  satisfies  the above conditions and  hence defines a homomorphism  $\rho\hskip 1.00006pt(\hskip 1.00006ptu\hskip 1.00006pt)\hskip 1.00006pt\colon\hskip 1.00006pt\pi_{\hskip 0.70004pt1}\hskip 1.00006pt(\hskip 1.49994ptX\hskip 0.50003pt,\hskip 1.99997ptb\hskip 1.49994pt)\hskip 1.99997pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 1.99997ptG$.   Since  elements  of  $C^{\hskip 0.70004pt0}\hskip 1.00006pt(\hskip 1.49994ptX\hskip 0.50003pt,\hskip 3.00003ptb\hskip 1.00006pt)$  are  equal  to $1$ at  $b$,  $\rho\hskip 1.00006pt(\hskip 1.00006ptu\hskip 1.00006pt)$ depends only  on  the cohomology  class $[\hskip 1.00006ptu\hskip 0.50003pt\hskip 1.00006pt]$,   and we get  a map

$$\quad\rho\hskip 1.99997pt\colon\hskip 1.99997ptH^{\hskip 0.70004pt1}\hskip 1.00006pt(\hskip 1.49994ptX\hskip 0.50003pt,\hskip 3.00003ptb\hskip 1.00006pt)\hskip 3.99994pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 3.99994pt\operatorname{Hom}\hskip 1.49994pt(\hskip 1.49994pt\pi_{\hskip 0.70004pt1}\hskip 1.00006pt(\hskip 1.49994ptX\hskip 0.50003pt,\hskip 1.99997ptb\hskip 1.49994pt)\hskip 0.50003pt,\hskip 3.00003ptG\hskip 1.49994pt)\hskip 3.00003pt.$$

In order  to construct  a map in  the opposite direction,   let  us  choose for every  point  $x\hskip 1.99997pt\in\hskip 1.99997ptX$  a path $s_{\hskip 0.35002ptx}$ such  that  $s_{\hskip 0.35002ptx}\hskip 1.49994pt(\hskip 1.00006pt0\hskip 1.00006pt)\hskip 3.99994pt=\hskip 3.99994ptb$ and  $s_{\hskip 0.35002ptx}\hskip 1.49994pt(\hskip 1.00006pt1\hskip 1.00006pt)\hskip 3.99994pt=\hskip 3.99994ptx$,   and choose as $s_{\hskip 0.70004ptb}$  the constant  path  with  the value $b$.   Let  us  define a map  $l\hskip 1.00006pt\colon\hskip 1.00006ptP\hskip 1.00006pt(\hskip 1.49994ptX\hskip 1.49994pt)\hskip 1.99997pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 1.99997ptL\hskip 1.49994pt(\hskip 1.49994ptX\hskip 0.50003pt,\hskip 1.99997ptb\hskip 1.49994pt)$  by  the rule

$$\quad l\hskip 1.49994pt(\hskip 1.00006ptp\hskip 1.49994pt)\hskip 3.99994pt=\hskip 3.99994pts_{\hskip 0.70004ptp\hskip 0.70004pt(\hskip 0.70004pt0\hskip 0.70004pt)}\hskip 1.00006pt\cdot\hskip 1.00006ptp\hskip 1.00006pt\cdot\hskip 3.00003pt\overline{s_{\hskip 0.70004ptp\hskip 0.70004pt(\hskip 0.70004pt1\hskip 0.70004pt)}}\hskip 3.99994pt.$$

If  $p\hskip 1.99997pt\sim\hskip 1.99997ptq$,   then,   in  particular ,  $p\hskip 1.00006pt(\hskip 1.00006pt0\hskip 1.00006pt)\hskip 3.99994pt=\hskip 3.99994ptq\hskip 1.00006pt(\hskip 1.00006pt0\hskip 1.00006pt)$ and  $p\hskip 1.00006pt(\hskip 1.00006pt1\hskip 1.00006pt)\hskip 3.99994pt=\hskip 3.99994ptq\hskip 1.00006pt(\hskip 1.00006pt1\hskip 1.00006pt)$ and  hence  $l\hskip 1.49994pt(\hskip 1.00006ptp\hskip 1.49994pt)\hskip 3.99994pt\sim\hskip 3.99994ptl\hskip 1.49994pt(\hskip 1.00006ptq\hskip 1.49994pt)$.   If  $p$  is  a constant  path,   then  $p\hskip 1.00006pt(\hskip 1.00006pt0\hskip 1.00006pt)\hskip 3.99994pt=\hskip 3.99994ptp\hskip 1.00006pt(\hskip 1.00006pt1\hskip 1.00006pt)$  and  $l\hskip 1.49994pt(\hskip 1.00006ptp\hskip 1.49994pt)$  is  homotopic  to  the constant  loop.   If  the product  $p\hskip 1.00006pt\cdot\hskip 1.00006ptq$  is  defined,   then $p\hskip 1.00006pt(\hskip 1.00006pt1\hskip 1.00006pt)\hskip 3.99994pt=\hskip 3.99994ptq\hskip 1.00006pt(\hskip 1.00006pt0\hskip 1.00006pt)$ and  hence  $\overline{s_{\hskip 0.70004ptp\hskip 0.70004pt(\hskip 0.70004pt1\hskip 0.70004pt)}}\hskip 1.99997pt\cdot\hskip 1.00006pts_{\hskip 0.70004ptq\hskip 0.70004pt(\hskip 0.70004pt0\hskip 0.70004pt)}\hskip 3.99994pt=\hskip 3.99994pt\overline{s_{\hskip 0.70004ptp\hskip 0.70004pt(\hskip 0.70004pt1\hskip 0.70004pt)}}\hskip 1.99997pt\cdot\hskip 1.00006pts_{\hskip 0.70004ptp\hskip 0.70004pt(\hskip 0.70004pt1\hskip 0.70004pt)}$  is  homotopic  to a constant  path.   It  follows  that  $l\hskip 1.49994pt(\hskip 1.00006ptp\hskip 1.00006pt\cdot\hskip 1.00006ptq\hskip 1.49994pt)\hskip 3.99994pt\sim\hskip 3.99994ptl\hskip 1.49994pt(\hskip 1.00006ptp\hskip 1.49994pt)\hskip 1.00006pt\cdot\hskip 1.00006ptl\hskip 1.49994pt(\hskip 1.00006ptq\hskip 1.49994pt)$.

Let  $h\hskip 1.00006pt\colon\hskip 1.00006ptL\hskip 1.49994pt(\hskip 1.49994ptX\hskip 0.50003pt,\hskip 1.99997ptb\hskip 1.49994pt)\hskip 1.99997pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 1.99997ptG$  be  a map satisfying  the above conditions.   Then  the properties of  $l$  imply  that  the map  $h^{\hskip 0.70004pt\sim}\hskip 1.00006pt\colon\hskip 1.00006ptP\hskip 1.00006pt(\hskip 1.49994ptX\hskip 1.49994pt)\hskip 1.99997pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 1.99997ptG$  defined  by  $h^{\hskip 0.70004pt\sim}\hskip 1.00006pt(\hskip 1.00006ptp\hskip 1.49994pt)\hskip 3.99994pt=\hskip 3.99994pth\hskip 1.49994pt(\hskip 1.49994ptl\hskip 1.49994pt(\hskip 1.00006ptp\hskip 1.49994pt)\hskip 1.49994pt)$  is  a cocycle.   Taking  the cohomology  classes of  the cocycles  $h^{\hskip 0.70004pt\sim}$  leads  to a map

$$\quad\varepsilon\hskip 1.99997pt\colon\hskip 1.99997pt\operatorname{Hom}\hskip 1.49994pt(\hskip 1.49994pt\pi_{\hskip 0.70004pt1}\hskip 1.00006pt(\hskip 1.49994ptX\hskip 0.50003pt,\hskip 1.99997ptb\hskip 1.49994pt)\hskip 0.50003pt,\hskip 3.00003ptG\hskip 1.49994pt)\hskip 3.99994pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 3.99994ptH^{\hskip 0.70004pt1}\hskip 1.00006pt(\hskip 1.49994ptX\hskip 0.50003pt,\hskip 3.00003ptb\hskip 1.00006pt)\hskip 3.00003pt.$$

2.1. Lemma.   Both maps  $\rho$  and  $\varepsilon$  are bijections and  $\varepsilon\hskip 3.99994pt=\hskip 3.99994pt\rho^{\hskip 0.70004pt-\hskip 0.70004pt1}$.   

Proof.   Since  $s_{\hskip 0.70004ptb}$  is  the constant  path,  $l\hskip 1.49994pt(\hskip 1.00006ptp\hskip 1.49994pt)\hskip 3.99994pt=\hskip 3.99994pts_{\hskip 0.70004ptb}\hskip 1.00006pt\cdot\hskip 1.00006ptp\hskip 1.00006pt\cdot\hskip 1.99997pt\overline{s_{\hskip 0.70004ptb}}\hskip 3.99994pt\hskip 3.99994pt\sim\hskip 3.99994pt\hskip 1.99997ptp$  for every  $p\hskip 1.99997pt\in\hskip 1.99997ptL\hskip 1.49994pt(\hskip 1.49994ptX\hskip 0.50003pt,\hskip 1.99997ptb\hskip 1.49994pt)$  and  hence  the restriction of  $h^{\hskip 0.70004pt\sim}$  to  $L\hskip 1.49994pt(\hskip 1.49994ptX\hskip 0.50003pt,\hskip 1.99997ptb\hskip 1.49994pt)$  is  equal  to  $h$.   Therefore  $\rho\hskip 1.00006pt\circ\hskip 1.00006pt\varepsilon$  is  the identity  map.   It  remains  to show  that  $\varepsilon\hskip 1.00006pt\circ\hskip 1.00006pt\rho$  is  the identity  map.   Let  $u\hskip 1.99997pt\in\hskip 1.99997ptZ^{\hskip 0.70004pt1}\hskip 1.00006pt(\hskip 1.49994ptX\hskip 0.50003pt,\hskip 3.00003ptb\hskip 1.00006pt)$.   Let  us consider  its restriction  $h$  to  $L\hskip 1.49994pt(\hskip 1.49994ptX\hskip 0.50003pt,\hskip 1.99997ptb\hskip 1.49994pt)$  and  the corresponding cocycle  $h^{\hskip 0.70004pt\sim}$.   If  $p\hskip 1.99997pt\in\hskip 1.99997ptP\hskip 1.00006pt(\hskip 1.49994ptX\hskip 1.49994pt)$,   then

$$\quad h^{\hskip 0.70004pt\sim}\hskip 1.00006pt(\hskip 1.00006ptp\hskip 1.49994pt)\hskip 3.99994pt=\hskip 3.99994pth\hskip 1.49994pt(\hskip 1.49994ptl\hskip 1.49994pt(\hskip 1.00006ptp\hskip 1.49994pt)\hskip 1.49994pt)\hskip 3.99994pt=\hskip 3.99994ptu\hskip 1.49994pt(\hskip 1.49994ptl\hskip 1.49994pt(\hskip 1.00006ptp\hskip 1.49994pt)\hskip 1.49994pt)$$

$$\quad\phantom{h^{\hskip 0.70004pt\sim}\hskip 1.00006pt(\hskip 1.00006ptp\hskip 1.49994pt)\hskip 3.99994pt=\hskip 3.99994pth\hskip 1.49994pt(\hskip 1.49994ptl\hskip 1.49994pt(\hskip 1.00006ptp\hskip 1.49994pt)\hskip 1.49994pt)\hskip 3.99994pt}=\hskip 3.99994ptu\hskip 1.49994pt\left(\hskip 1.99997pts_{\hskip 0.70004ptp\hskip 0.70004pt(\hskip 0.70004pt0\hskip 0.70004pt)}\hskip 1.00006pt\cdot\hskip 1.00006ptp\hskip 1.00006pt\cdot\hskip 3.00003pt\overline{s_{\hskip 0.70004ptp\hskip 0.70004pt(\hskip 0.70004pt1\hskip 0.70004pt)}}\hskip 3.00003pt\right)$$

$$\quad\phantom{h^{\hskip 0.70004pt\sim}\hskip 1.00006pt(\hskip 1.00006ptp\hskip 1.49994pt)\hskip 3.99994pt=\hskip 3.99994pth\hskip 1.49994pt(\hskip 1.49994ptl\hskip 1.49994pt(\hskip 1.00006ptp\hskip 1.49994pt)\hskip 1.49994pt)\hskip 3.99994pt}=\hskip 3.99994ptu\hskip 1.49994pt\left(\hskip 1.00006pts_{\hskip 0.70004ptp\hskip 0.70004pt(\hskip 0.70004pt0\hskip 0.70004pt)}\hskip 1.49994pt\right)\hskip 1.00006pt\cdot\hskip 1.00006ptu\hskip 1.49994pt\left(\hskip 1.00006ptp\hskip 1.49994pt\right)\hskip 1.00006pt\cdot\hskip 3.00003ptu\hskip 1.49994pt\left(\hskip 3.00003pt\overline{s_{\hskip 0.70004ptp\hskip 0.70004pt(\hskip 0.70004pt1\hskip 0.70004pt)}}\hskip 3.00003pt\right)$$

$$\quad\phantom{h^{\hskip 0.70004pt\sim}\hskip 1.00006pt(\hskip 1.00006ptp\hskip 1.49994pt)\hskip 3.99994pt=\hskip 3.99994pth\hskip 1.49994pt(\hskip 1.49994ptl\hskip 1.49994pt(\hskip 1.00006ptp\hskip 1.49994pt)\hskip 1.49994pt)\hskip 3.99994pt}=\hskip 3.99994ptu\hskip 1.49994pt\left(\hskip 1.00006pts_{\hskip 0.70004ptp\hskip 0.70004pt(\hskip 0.70004pt0\hskip 0.70004pt)}\hskip 1.49994pt\right)\hskip 1.00006pt\cdot\hskip 1.00006ptu\hskip 1.49994pt\left(\hskip 1.00006ptp\hskip 1.49994pt\right)\hskip 1.00006pt\cdot\hskip 3.00003ptu\hskip 1.49994pt\left(\hskip 3.00003pts_{\hskip 0.70004ptp\hskip 0.70004pt(\hskip 0.70004pt1\hskip 0.70004pt)}\hskip 3.00003pt\right)^{\hskip 0.70004pt-\hskip 0.70004pt1}\hskip 3.00003pt.$$

It  follows  that  $h^{\hskip 0.70004pt\sim}\hskip 3.99994pt=\hskip 3.99994ptc\hskip 1.00006pt\bullet\hskip 1.00006ptu$,   where  $c$  is  the $0$-cochain  $x\hskip 3.99994pt\longmapsto\hskip 3.99994ptu\hskip 1.49994pt\left(\hskip 1.00006pts_{\hskip 0.70004ptx}\hskip 1.49994pt\right)$.   Since  $s_{\hskip 0.70004ptb}$  is  the constant  path with  the value  $b$  and  $u\hskip 1.99997pt\in\hskip 1.99997ptZ^{\hskip 0.70004pt1}\hskip 1.00006pt(\hskip 1.49994ptX\hskip 0.50003pt,\hskip 3.00003ptb\hskip 1.00006pt)$,   the $0$-cochain $c$  belongs  to  $C^{\hskip 0.70004pt0}\hskip 1.00006pt(\hskip 1.49994ptX\hskip 0.50003pt,\hskip 3.00003ptb\hskip 1.00006pt)$.   Therefore  the cohomology  class of  $h^{\hskip 0.70004pt\sim}$  is  equal  to  the cohomology  class of  $u$.   Since $u$  was an arbitrary  cocycle,   this implies  that  $\varepsilon\hskip 1.00006pt\circ\hskip 1.00006pt\rho$  is  the identity  map.    $\blacksquare$

2.2. Theorem.   The maps  $\rho$  and  $\varepsilon$  are canonical  bijections  between  the sets  $H^{\hskip 0.70004pt1}\hskip 1.00006pt(\hskip 1.49994ptX\hskip 0.50003pt,\hskip 3.00003ptb\hskip 1.00006pt)$  and  $\operatorname{Hom}\hskip 1.49994pt(\hskip 1.49994pt\pi_{\hskip 0.70004pt1}\hskip 1.00006pt(\hskip 1.49994ptX\hskip 0.50003pt,\hskip 1.99997ptb\hskip 1.49994pt)\hskip 0.50003pt,\hskip 3.00003ptG\hskip 1.49994pt)$.   In  particular ,  $\varepsilon$  does not  depend  on  the choice of  paths  $s_{\hskip 0.70004ptx}$.   

Proof.   It  remains  only  to check  that  $\varepsilon$  does not  depend on  the choice of  paths  $s_{\hskip 0.70004ptx}$.   This follows from  the fact  that  definition of  $\rho$  does not  involve any  choices.    $\blacksquare$

3. Short  cochains

Short  paths and  short  homotopies.   Let  us  fix an open covering $\mathcal{U}$ of  $X$.   Let  call  a path $p$ in $X$  short  if  $p\hskip 1.00006pt(\hskip 1.49994ptI\hskip 1.49994pt)\hskip 1.99997pt\subset\hskip 1.99997ptU$  for some $U\hskip 1.99997pt\in\hskip 1.99997pt\mathcal{U}$.   A  homotopy  $I\hskip 1.00006pt\times\hskip 1.00006pt[\hskip 1.00006pt0\hskip 0.50003pt,\hskip 1.99997pt1\hskip 1.00006pt]\hskip 1.99997pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 1.99997ptX$  is  called  short  if  its image  is  contained  in  $U$  for some $U\hskip 1.99997pt\in\hskip 1.99997pt\mathcal{U}$.   Obviously ,   a short  homotopy  is  a  homotopy  between  two short  paths.   One can  replace in  the definitions of  cochains and cocycles arbitrary  paths by  short  paths and  arbitrary  homotopies  by  short  homotopies.

In more details,   let  $P_{\hskip 0.70004pt\mathcal{U}}\hskip 1.00006pt(\hskip 1.49994ptX\hskip 1.49994pt)$ be  the set  of  short  paths in $X$.   Every $0$-cochain  should  be considered as  short  because every  point  belongs  to some $U\hskip 1.99997pt\in\hskip 1.99997pt\mathcal{U}$.   A  short  $1$-cochain  of  the pair $(\hskip 1.49994ptX\hskip 0.50003pt,\hskip 3.00003ptY\hskip 1.49994pt)$  is  a map  $u\hskip 1.00006pt\colon\hskip 1.00006ptP_{\hskip 0.70004pt\mathcal{U}}\hskip 1.00006pt(\hskip 1.49994ptX\hskip 1.49994pt)\hskip 1.99997pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 1.99997ptG$  such  that  $u\hskip 1.49994pt(\hskip 1.00006ptp\hskip 1.49994pt)\hskip 3.99994pt=\hskip 3.99994pt1$  if  $p\hskip 1.99997pt\in\hskip 1.99997ptP\hskip 1.00006pt(\hskip 1.49994ptY\hskip 1.49994pt)$.   The set  of  all  short  $n$-cochains,   where $n\hskip 3.99994pt=\hskip 3.99994pt0$  or  $1$,   is  denoted  by  $C^{\hskip 0.70004ptn}_{\hskip 0.70004pt\mathcal{U}}\hskip 1.00006pt(\hskip 1.49994ptX\hskip 0.50003pt,\hskip 3.00003ptY\hskip 1.49994pt)$.   A short  $1$-cochain $u$ is  called a short  cocycle  if  $u\hskip 1.49994pt(\hskip 1.00006ptp\hskip 1.49994pt)\hskip 3.99994pt=\hskip 3.99994ptu\hskip 1.49994pt(\hskip 1.00006ptq\hskip 1.49994pt)$  when  there exists a short  homotopy  relatively  to  $\{\hskip 1.99997pt0\hskip 0.50003pt,\hskip 1.99997pt1\hskip 1.49994pt\}$  between $p$ and $q$,   and  also  $u\hskip 1.49994pt(\hskip 1.00006ptp\hskip 1.00006pt\cdot\hskip 1.00006ptq\hskip 1.49994pt)\hskip 3.99994pt=\hskip 3.99994ptu\hskip 1.49994pt(\hskip 1.00006ptp\hskip 1.49994pt)\hskip 1.00006pt\cdot\hskip 1.00006ptu\hskip 1.49994pt(\hskip 1.00006ptq\hskip 1.49994pt)$  when  the product  $p\hskip 1.00006pt\cdot\hskip 1.00006ptq$  is  defined and  the path $p\hskip 1.00006pt\cdot\hskip 1.00006ptq$  is  short.   The set  of  all  short  cocycles  is  denoted  by  $Z^{\hskip 0.70004pt1}_{\hskip 0.70004pt\mathcal{U}}\hskip 1.00006pt(\hskip 1.49994ptX\hskip 0.50003pt,\hskip 3.00003ptY\hskip 1.49994pt)$.

The action of  $0$-cochains on  short $1$-cochains  is  defined exactly  as before,   as also  the relation of  being  cohomologous  cocycles.   Being cohomologous  is  an equivalence relation,   and  the set  of  equivalence classes of  short $1$-cocycles  is  denoted  by  $H^{\hskip 0.70004pt1}_{\hskip 0.70004pt\mathcal{U}}\hskip 1.00006pt(\hskip 1.49994ptX\hskip 0.50003pt,\hskip 3.00003ptY\hskip 1.49994pt)$.   Restricting  maps $P\hskip 1.00006pt(\hskip 1.49994ptX\hskip 1.49994pt)\hskip 1.99997pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 1.99997ptG$ to $P_{\hskip 0.70004pt\mathcal{U}}\hskip 1.00006pt(\hskip 1.49994ptX\hskip 1.49994pt)$  leads  to maps  $F^{\hskip 0.70004pt1}\hskip 1.00006pt(\hskip 1.49994ptX\hskip 0.50003pt,\hskip 3.00003ptY\hskip 1.49994pt)\hskip 3.99994pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 3.99994ptF^{\hskip 0.70004pt1}_{\hskip 0.70004pt\mathcal{U}}\hskip 1.00006pt(\hskip 1.49994ptX\hskip 0.50003pt,\hskip 3.00003ptY\hskip 1.49994pt)$,   where  $F\hskip 3.99994pt=\hskip 3.99994ptC$,  $Z$,  or  $H$.

3.1. Theorem.   For  $F\hskip 3.99994pt=\hskip 3.99994ptZ$  or  $H$  the map  $F^{\hskip 0.70004pt1}\hskip 1.00006pt(\hskip 1.49994ptX\hskip 0.50003pt,\hskip 3.00003ptY\hskip 1.49994pt)\hskip 3.99994pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 3.99994ptF^{\hskip 0.70004pt1}_{\hskip 0.70004pt\mathcal{U}}\hskip 1.00006pt(\hskip 1.49994ptX\hskip 0.50003pt,\hskip 3.00003ptY\hskip 1.49994pt)$  is  a  bijection.   

Proof.   Let  us  deal  with cocycles first .   Given a short  cocycle  $u$,   let  us define a cochain  $u^{\hskip 0.70004pt\sharp}$  as follows.   Let  $p\hskip 1.00006pt\colon\hskip 1.00006ptI\hskip 1.99997pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 1.99997ptX$  be a path.   By  Lebesgue  lemma  there exists a subdivision of  the interval $I$ into subintervals such  that $p$ maps each of  these subintervals into some $U\hskip 1.99997pt\in\hskip 1.99997pt\mathcal{U}$.   In other  words,   there are numbers  $0\hskip 3.99994pt=\hskip 3.99994pta_{\hskip 0.70004pt1}\hskip 3.99994pt<\hskip 3.99994pta_{\hskip 0.70004pt2}\hskip 3.99994pt<\hskip 3.99994pt\ldots\hskip 3.99994pt<\hskip 3.99994pta_{\hskip 0.70004ptn}\hskip 3.99994pt=\hskip 3.99994pt1$  such  that  $p\hskip 1.49994pt\hskip 1.49994pt[\hskip 1.00006pta_{\hskip 0.70004pti\hskip 0.70004pt-\hskip 0.70004pt1}\hskip 1.00006pt,\hskip 3.00003pta_{\hskip 0.70004pti}\hskip 1.00006pt]\hskip 1.49994pt)$  is  contained  in  some $U_{\hskip 0.70004pti}\hskip 1.99997pt\in\hskip 1.99997pt\mathcal{U}$ for each  $i\hskip 3.99994pt=\hskip 3.99994pt1\hskip 0.50003pt,\hskip 3.00003pt2\hskip 0.50003pt,\hskip 3.00003pt\ldots\hskip 0.50003pt,\hskip 3.00003ptn$.   For each such number  $i$  let  $p_{\hskip 0.70004pti}\hskip 3.99994pt=\hskip 3.99994ptp\hskip 1.00006pt\circ\hskip 1.00006pt\varphi_{\hskip 0.70004pti}$,   where  $\varphi_{\hskip 0.70004pti}\hskip 1.00006pt\colon\hskip 1.00006pt[\hskip 1.00006pt0\hskip 0.50003pt,\hskip 1.99997pt1\hskip 1.00006pt]\hskip 1.99997pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 1.99997pt[\hskip 1.49994pta_{\hskip 0.70004pti\hskip 0.70004pt-\hskip 0.70004pt1}\hskip 1.00006pt,\hskip 3.00003pta_{\hskip 0.70004pti}\hskip 1.99997pt]$  is  an  increasing  homeomorphism.   Then each  $p_{\hskip 0.70004pti}$  is  a  short  path and  hence  $u\hskip 1.49994pt(\hskip 1.00006ptp_{\hskip 0.70004pti}\hskip 1.49994pt)$  is  defined.   Let

(2)

$$\quad u^{\hskip 0.70004pt\sharp}\hskip 1.00006pt(\hskip 1.00006ptp\hskip 1.49994pt)\hskip 3.99994pt=\hskip 3.99994ptu\hskip 1.49994pt(\hskip 1.00006ptp_{\hskip 0.70004pt1}\hskip 1.49994pt)\hskip 1.00006pt\cdot\hskip 1.00006ptu\hskip 1.49994pt(\hskip 1.00006ptp_{\hskip 0.70004pt2}\hskip 1.49994pt)\hskip 1.00006pt\cdot\hskip 1.00006pt\ldots\hskip 1.00006pt\cdot\hskip 1.00006ptu\hskip 1.49994pt(\hskip 1.00006ptp_{\hskip 0.70004ptn}\hskip 1.49994pt)\hskip 3.00003pt.$$

Let  us  check  that  $u^{\hskip 0.70004pt\sharp}\hskip 1.00006pt(\hskip 1.00006ptp\hskip 1.49994pt)$  is  independent  from  the choices involved.   To begin  with,   for each  $i$  different  choices of  $\varphi_{\hskip 0.70004pti}$  lead  to paths differing  by  a  reparametrization.   Since  $p_{\hskip 0.70004pti}$  is  a  short  path,   a reparametrization of  $p_{\hskip 0.70004pti}$  is  homotopic  to  $p_{\hskip 0.70004pti}$  by  a  short  homotopy .   Since $u$ is  a  short $1$-cocycle,   this implies  that $u\hskip 1.49994pt(\hskip 1.00006ptp_{\hskip 0.70004pti}\hskip 1.49994pt)$ does not  depend on  the choice of  $\varphi_{\hskip 0.70004pti}$.   It  follows  that $u^{\hskip 0.70004pt\sharp}\hskip 1.00006pt(\hskip 1.00006ptp\hskip 1.49994pt)$ does not  depend on  the choice of  homeomorphisms  $\varphi_{\hskip 0.70004pti}$.   Replacing  $[\hskip 1.49994pta_{\hskip 0.70004pti\hskip 0.70004pt-\hskip 0.70004pt1}\hskip 1.00006pt,\hskip 3.00003pta_{\hskip 0.70004pti}\hskip 1.99997pt]$  by  its subdivisions  will  not  change  $u^{\hskip 0.70004pt\sharp}\hskip 1.00006pt(\hskip 1.00006ptp\hskip 1.49994pt)$  because $u$  is  a short  cocycle.   Since every  two subdivisions of  $I\hskip 3.99994pt=\hskip 3.99994pt[\hskip 1.00006pt0\hskip 0.50003pt,\hskip 1.99997pt1\hskip 1.00006pt]$  have a common subdivision,   it  follows  that  $u^{\hskip 0.70004pt\sharp}\hskip 1.00006pt(\hskip 1.00006ptp\hskip 1.49994pt)$  does not  depend on  the subdivision used.   Therefore  $u^{\hskip 0.70004pt\sharp}$  is  correctly  defined.

Let  us  prove  that  $u^{\hskip 0.70004pt\sharp}$  is  a cocycle of  $(\hskip 1.49994ptX\hskip 0.50003pt,\hskip 3.00003ptY\hskip 1.49994pt)$.   Clearly ,  $u^{\hskip 0.70004pt\sharp}\hskip 1.00006pt(\hskip 1.00006ptp\hskip 1.49994pt)\hskip 3.99994pt=\hskip 3.99994pt1$  if  $p\hskip 1.99997pt\in\hskip 1.99997ptP\hskip 1.00006pt(\hskip 1.49994ptY\hskip 1.49994pt)$.   The fact  that  $u^{\hskip 0.70004pt\sharp}\hskip 1.00006pt(\hskip 1.00006ptp\hskip 1.00006pt\cdot\hskip 1.00006ptq\hskip 1.49994pt)\hskip 3.99994pt=\hskip 3.99994ptu^{\hskip 0.70004pt\sharp}\hskip 1.00006pt(\hskip 1.00006ptp\hskip 1.49994pt)\hskip 1.00006pt\cdot\hskip 1.00006ptu^{\hskip 0.70004pt\sharp}\hskip 1.00006pt(\hskip 1.00006ptq\hskip 1.49994pt)$  when  $p\hskip 1.00006pt\cdot\hskip 1.00006ptq$  is  defined  follows  from  the independence of  $u^{\hskip 0.70004pt\sharp}\hskip 1.00006pt(\hskip 1.00006ptp\hskip 1.00006pt\cdot\hskip 1.00006ptq\hskip 1.49994pt)$  on  the subdivision used.   Let  us  prove  that  $u^{\hskip 0.70004pt\sharp}\hskip 1.00006pt(\hskip 1.00006ptp\hskip 1.49994pt)\hskip 1.99997pt=\hskip 1.99997ptu^{\hskip 0.70004pt\sharp}\hskip 1.00006pt(\hskip 1.00006ptq\hskip 1.49994pt)$ when  $p\hskip 3.99994pt\sim\hskip 3.99994ptq$.   Let $h\hskip 1.00006pt\colon\hskip 1.00006ptI\hskip 1.00006pt\times\hskip 1.00006pt[\hskip 1.00006pt0\hskip 0.50003pt,\hskip 1.99997pt1\hskip 1.00006pt]\hskip 1.99997pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 1.99997ptX$  be a  homotopy  between  $p$ and $q$ such  that  $h\hskip 1.49994pt(\hskip 1.00006pt0\hskip 0.50003pt,\hskip 1.99997ptt\hskip 1.49994pt)\hskip 3.99994pt=\hskip 3.99994ptp\hskip 1.00006pt(\hskip 1.00006pt0\hskip 1.49994pt)\hskip 3.99994pt=\hskip 3.99994ptq\hskip 1.00006pt(\hskip 1.00006pt0\hskip 1.49994pt)$ and  $h\hskip 1.49994pt(\hskip 1.00006pt1\hskip 0.50003pt,\hskip 1.99997ptt\hskip 1.49994pt)\hskip 3.99994pt=\hskip 3.99994ptp\hskip 1.00006pt(\hskip 1.00006pt1\hskip 1.49994pt)\hskip 3.99994pt=\hskip 3.99994ptq\hskip 1.00006pt(\hskip 1.00006pt1\hskip 1.49994pt)$  for every  $t\hskip 1.99997pt\in\hskip 1.99997pt[\hskip 1.00006pt0\hskip 0.50003pt,\hskip 1.99997pt1\hskip 1.00006pt]$.

By  Lebesgue  lemma  there exists a natural  number $n$ such  that  $h$  maps every  square

$$\quad\left[\hskip 1.99997pt\frac{\hskip 1.00006pti\hskip 1.00006pt}{n}\hskip 1.99997pt,\hskip 3.99994pt\frac{i\hskip 1.99997pt+\hskip 1.99997pt1}{n}\hskip 1.99997pt\right]\hskip 1.99997pt\times\hskip 1.99997pt\left[\hskip 1.99997pt\frac{\hskip 1.00006ptk\hskip 1.00006pt}{n}\hskip 1.99997pt,\hskip 3.99994pt\frac{k\hskip 1.99997pt+\hskip 1.99997pt1}{n}\hskip 1.99997pt\right]\hskip 3.00003pt$$

with  $i\hskip 0.50003pt,\hskip 3.00003ptk\hskip 3.99994pt=\hskip 3.99994pt0\hskip 0.50003pt,\hskip 3.00003pt1\hskip 0.50003pt,\hskip 3.00003pt\ldots\hskip 0.50003pt,\hskip 3.00003ptn\hskip 1.99997pt-\hskip 1.99997pt1$  into some  $U\hskip 1.99997pt\in\hskip 1.99997pt\mathcal{U}$.   Let  us consider segments of  the form

$$\quad K\hskip 3.99994pt=\hskip 3.99994pt\left[\hskip 1.99997pt\frac{\hskip 1.00006ptk\hskip 1.00006pt}{n}\hskip 1.99997pt,\hskip 3.99994pt\frac{k\hskip 1.99997pt+\hskip 1.99997pt1}{n}\hskip 1.99997pt\right]\hskip 1.99997pt\times\hskip 1.99997pt\frac{\hskip 1.00006pti\hskip 1.00006pt}{n}\quad\ \mbox{or}\quad\ \frac{\hskip 1.00006pti\hskip 1.00006pt}{n}\hskip 1.99997pt\times\hskip 1.99997pt\left[\hskip 1.99997pt\frac{\hskip 1.00006ptk\hskip 1.00006pt}{n}\hskip 1.99997pt,\hskip 3.99994pt\frac{k\hskip 1.99997pt+\hskip 1.99997pt1}{n}\hskip 1.99997pt\right]$$

with  $0\hskip 1.99997pt\leqslant\hskip 1.99997pti\hskip 1.99997pt\leqslant\hskip 1.99997ptn$  and  $0\hskip 1.99997pt\leqslant\hskip 1.99997ptk\hskip 1.99997pt\leqslant\hskip 1.99997ptn\hskip 1.99997pt-\hskip 1.99997pt1$.   These segments are nothing else but  the sides of  the above squares.   For each such segment  $K$  let  $\varphi_{\hskip 1.04996ptK}\hskip 1.00006pt\colon\hskip 1.00006ptI\hskip 1.99997pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 1.99997pt\ I\hskip 1.00006pt\times\hskip 1.00006ptI$  be  the path defined  by

$$\quad\varphi_{\hskip 1.04996ptK}\hskip 1.00006pt(\hskip 1.00006pts\hskip 1.49994pt)\hskip 3.99994pt=\hskip 3.99994pt\left(\hskip 1.99997pt\frac{\hskip 1.00006pti\hskip 1.00006pt}{n}\hskip 1.00006pt,\hskip 3.99994pt\frac{k\hskip 1.99997pt+\hskip 1.99997pts}{n}\hskip 1.99997pt\right)\quad\ \mbox{or}\quad\ \left(\hskip 1.99997pt\frac{k\hskip 1.99997pt+\hskip 1.99997pts}{n}\hskip 1.00006pt,\hskip 3.99994pt\frac{\hskip 1.00006pti\hskip 1.00006pt}{n}\hskip 1.99997pt\right)$$

respectively .   Then  $\varphi_{\hskip 1.04996ptK}\hskip 1.00006pt(\hskip 1.49994ptI\hskip 1.49994pt)\hskip 3.99994pt=\hskip 3.99994ptK$  and  $h_{\hskip 1.04996ptK}\hskip 3.99994pt=\hskip 3.99994pth\hskip 1.00006pt\circ\hskip 1.49994pt\varphi_{\hskip 0.70004ptK}$  is  a  short  path.   If  $K$  is  contained  in  either  $0\hskip 1.00006pt\times\hskip 1.00006pt[\hskip 1.00006pt0\hskip 0.50003pt,\hskip 1.99997pt1\hskip 1.00006pt]$  or  $1\hskip 1.00006pt\times\hskip 1.00006pt[\hskip 1.00006pt0\hskip 0.50003pt,\hskip 1.99997pt1\hskip 1.00006pt]$,   then  $h_{\hskip 1.04996ptK}$  is  a  constant  path.

Let  $p_{\hskip 0.35002ptj}$,   where $j\hskip 3.99994pt=\hskip 3.99994pt0\hskip 0.50003pt,\hskip 3.00003pt1\hskip 0.50003pt,\hskip 3.00003pt\ldots\hskip 0.50003pt,\hskip 3.00003ptn$,   be  the path in  $X$  defined  by  $p_{\hskip 0.35002ptj}\hskip 1.00006pt(\hskip 1.00006pts\hskip 1.49994pt)\hskip 3.99994pt=\hskip 3.99994pth\hskip 1.49994pt(\hskip 1.00006pts\hskip 0.50003pt,\hskip 3.00003ptj/n\hskip 1.49994pt)$.   Then  $p_{\hskip 0.70004pt0}\hskip 3.99994pt=\hskip 3.99994ptp$,  $p_{\hskip 0.70004ptn}\hskip 3.99994pt=\hskip 3.99994ptq$,   and  hence  it  is  sufficient  to prove  that  $u^{\hskip 0.70004pt\sharp}\hskip 1.00006pt(\hskip 1.00006ptp_{\hskip 0.35002ptj}\hskip 1.49994pt)\hskip 3.99994pt=\hskip 3.99994ptu^{\hskip 0.70004pt\sharp}\hskip 1.00006pt(\hskip 1.00006ptp_{\hskip 0.35002ptj\hskip 0.70004pt-\hskip 0.70004pt1}\hskip 1.49994pt)$  for every  $j\hskip 3.99994pt=\hskip 3.99994pt1\hskip 0.50003pt,\hskip 3.00003pt2\hskip 0.50003pt,\hskip 3.00003pt\ldots\hskip 0.50003pt,\hskip 3.00003ptn$.   Let  us consider  the rectangle  $Q_{\hskip 1.04996ptj}\hskip 3.99994pt=\hskip 3.99994ptI\hskip 1.00006pt\times\hskip 1.00006pt[\hskip 1.00006pt(\hskip 1.00006ptj\hskip 1.99997pt-\hskip 1.99997pt1\hskip 1.00006pt)/n\hskip 0.50003pt,\hskip 3.00003ptj/n\hskip 1.00006pt]$.

Let  $K_{\hskip 0.70004pt1}\hskip 1.00006pt,\hskip 3.99994ptK_{\hskip 1.04996pt2}\hskip 1.00006pt,\hskip 3.99994pt\ldots\hskip 1.00006pt,\hskip 3.99994ptK_{\hskip 0.70004ptn}$  be  the segments of  the above form contained  in  $I\hskip 1.00006pt\times\hskip 1.00006ptj/n$  and  listed  from  left  to  right ,   and  let  $k_{\hskip 0.70004pti}\hskip 3.99994pt=\hskip 3.99994pth_{\hskip 1.04996ptK_{\hskip 0.50003pti}}$.   Similarly ,   let  $L_{\hskip 1.04996pt1}\hskip 1.00006pt,\hskip 3.99994ptL_{\hskip 1.39998pt2}\hskip 1.00006pt,\hskip 3.99994pt\ldots\hskip 1.00006pt,\hskip 3.99994ptL_{\hskip 0.70004ptn}$  be  the segments of  the above form contained  in  $I\hskip 1.00006pt\times\hskip 1.00006pt(\hskip 1.00006ptj\hskip 1.99997pt-\hskip 1.99997pt1\hskip 1.00006pt)\hskip 1.00006pt/n$  and  listed  from  left  to  right ,   and  let  $l_{\hskip 0.70004pti}\hskip 3.99994pt=\hskip 3.99994pth_{\hskip 1.04996ptL_{\hskip 0.74997pti}}$.   Then every  $k_{\hskip 0.70004pti}$  and every  $l_{\hskip 0.70004pti}$  is  a  short  path and

$$\quad p_{\hskip 0.35002ptj}\hskip 3.99994pt=\hskip 3.99994ptk_{\hskip 0.70004pt1}\hskip 1.00006pt\cdot\hskip 1.99997ptk_{\hskip 1.39998pt2}\hskip 1.00006pt\cdot\hskip 1.99997pt\ldots\hskip 1.99997pt\cdot\hskip 1.99997ptk_{\hskip 0.70004ptn}\hskip 3.99994pt,\qquad p_{\hskip 0.35002ptj\hskip 0.70004pt-\hskip 0.70004pt1}\hskip 3.99994pt=\hskip 3.99994ptl_{\hskip 0.70004pt1}\hskip 1.00006pt\cdot\hskip 1.99997ptl_{\hskip 1.04996pt2}\hskip 1.00006pt\cdot\hskip 1.99997pt\ldots\hskip 1.99997pt\cdot\hskip 1.99997ptl_{\hskip 0.70004ptn}\hskip 3.99994pt.$$

Finally ,   let  $M_{\hskip 1.04996pt0}\hskip 1.00006pt,\hskip 3.99994ptM_{\hskip 0.70004pt1}\hskip 1.00006pt,\hskip 3.99994pt\ldots\hskip 1.00006pt,\hskip 3.99994ptM_{\hskip 0.70004ptn}$  be  the vertical  segments of  the above form contained  in  the rectangle  $Q_{\hskip 1.04996ptj}$  and  listed  from  left  to  right ,   and  let  $m_{\hskip 0.70004pti}\hskip 3.99994pt=\hskip 3.99994pth_{\hskip 1.39998ptM_{\hskip 0.50003pti}}$.   Then every  $m_{\hskip 0.70004pti}$  is  a  short  path and $m_{\hskip 1.39998pt0}\hskip 1.00006pt,\hskip 3.99994ptm_{\hskip 0.70004ptn}$ are constant  paths.   By  the definition,

$$\quad u^{\hskip 0.70004pt\sharp}\hskip 1.00006pt(\hskip 1.00006ptp_{\hskip 0.35002ptj}\hskip 1.49994pt)\hskip 3.99994pt=\hskip 3.99994ptu\hskip 1.49994pt(\hskip 1.00006ptk_{\hskip 0.70004pt1}\hskip 1.49994pt)\hskip 1.00006pt\cdot\hskip 1.00006ptu\hskip 1.49994pt(\hskip 1.00006ptk_{\hskip 1.39998pt2}\hskip 1.49994pt)\hskip 1.00006pt\cdot\hskip 1.00006pt\ldots\hskip 1.00006pt\cdot\hskip 1.00006ptu\hskip 1.49994pt(\hskip 1.00006ptk_{\hskip 0.70004ptn}\hskip 1.49994pt)\hskip 3.99994pt,\qquad u^{\hskip 0.70004pt\sharp}\hskip 1.00006pt(\hskip 1.00006ptp_{\hskip 0.35002ptj\hskip 0.70004pt-\hskip 0.70004pt1}\hskip 1.49994pt)\hskip 3.99994pt=\hskip 3.99994ptu\hskip 1.49994pt(\hskip 1.00006ptl_{\hskip 0.70004pt1}\hskip 1.49994pt)\hskip 1.00006pt\cdot\hskip 1.00006ptu\hskip 1.49994pt(\hskip 1.00006ptl_{\hskip 0.70004pt2}\hskip 1.49994pt)\hskip 1.00006pt\cdot\hskip 1.00006pt\ldots\hskip 1.00006pt\cdot\hskip 1.00006ptu\hskip 1.49994pt(\hskip 1.00006ptl_{\hskip 0.70004ptn}\hskip 1.49994pt)\hskip 3.00003pt.$$

As we pointed out  at  the beginning of  Section  Non-abelian  cohomology  and  Seifert–van Kampen  theorem,  if  $m$  is  a constant  path,   then  $u\hskip 1.49994pt(\hskip 1.00006ptm\hskip 1.49994pt)\hskip 3.99994pt=\hskip 3.99994pt1$  By  applying  this remark  to  $m_{\hskip 1.39998pt0}\hskip 1.00006pt,\hskip 3.99994ptm_{\hskip 0.70004ptn}$  we see  that

$$\quad u^{\hskip 0.70004pt\sharp}\hskip 1.00006pt(\hskip 1.00006ptp_{\hskip 0.35002ptj}\hskip 1.49994pt)\hskip 11.49995pt\hskip 3.99994pt=\hskip 3.99994ptu\hskip 1.49994pt(\hskip 1.00006ptm_{\hskip 1.39998pt0}\hskip 1.49994pt)\hskip 1.00006pt\cdot\hskip 1.00006ptu\hskip 1.49994pt(\hskip 1.00006ptk_{\hskip 0.70004pt1}\hskip 1.49994pt)\hskip 1.00006pt\cdot\hskip 1.00006ptu\hskip 1.49994pt(\hskip 1.00006ptk_{\hskip 1.39998pt2}\hskip 1.49994pt)\hskip 1.00006pt\cdot\hskip 1.00006pt\ldots\hskip 1.00006pt\cdot\hskip 1.00006ptu\hskip 1.49994pt(\hskip 1.00006ptk_{\hskip 0.70004ptn}\hskip 1.49994pt)\quad\ \mbox{and}\quad\$$

$$\quad u^{\hskip 0.70004pt\sharp}\hskip 1.00006pt(\hskip 1.00006ptp_{\hskip 0.35002ptj\hskip 0.70004pt-\hskip 0.70004pt1}\hskip 1.49994pt)\hskip 3.99994pt=\hskip 3.99994ptu\hskip 1.49994pt(\hskip 1.00006ptl_{\hskip 0.70004pt1}\hskip 1.49994pt)\hskip 1.00006pt\cdot\hskip 1.00006ptu\hskip 1.49994pt(\hskip 1.00006ptl_{\hskip 0.70004pt2}\hskip 1.49994pt)\hskip 1.00006pt\cdot\hskip 1.00006pt\ldots\hskip 1.00006pt\cdot\hskip 1.00006ptu\hskip 1.49994pt(\hskip 1.00006ptl_{\hskip 0.70004ptn}\hskip 1.49994pt)\hskip 1.00006pt\cdot\hskip 1.00006ptu\hskip 1.49994pt(\hskip 1.00006ptm_{\hskip 0.70004ptn}\hskip 1.49994pt)\hskip 3.00003pt.$$

Therefore,   it  is  sufficient  to prove  that  these  two products are equal.

We will  prove  that ,   moreover ,   all  products of  the form

$$\quad g_{\hskip 0.70004pti}\hskip 3.99994pt=\hskip 3.99994ptu\hskip 1.49994pt(\hskip 1.00006ptl_{\hskip 0.70004pt1}\hskip 1.49994pt)\hskip 1.00006pt\cdot\hskip 1.00006pt\ldots\hskip 1.00006pt\cdot\hskip 1.00006ptu\hskip 1.49994pt(\hskip 1.00006ptl_{\hskip 0.70004pti\hskip 0.70004pt-\hskip 0.70004pt1}\hskip 1.49994pt)\hskip 1.00006pt\cdot\hskip 1.00006ptu\hskip 1.49994pt(\hskip 1.00006ptm_{\hskip 0.70004pti}\hskip 1.49994pt)\hskip 1.00006pt\cdot\hskip 1.00006ptu\hskip 1.49994pt(\hskip 1.00006ptk_{\hskip 0.70004pti}\hskip 1.49994pt)\hskip 1.00006pt\cdot\hskip 1.00006pt\ldots\hskip 1.00006pt\cdot\hskip 1.00006ptu\hskip 1.49994pt(\hskip 1.00006ptk_{\hskip 0.70004ptn}\hskip 1.49994pt)$$

with  $i\hskip 3.99994pt=\hskip 3.99994pt0\hskip 0.50003pt,\hskip 3.00003pt1\hskip 0.50003pt,\hskip 3.00003pt\ldots\hskip 0.50003pt,\hskip 3.00003ptn$  are equal.

It  is  sufficient  to prove  that  $g_{\hskip 0.70004pti}\hskip 3.99994pt=\hskip 3.99994ptg_{\hskip 0.70004pti\hskip 0.70004pt+\hskip 0.70004pt1}$  for  $i\hskip 1.99997pt\leqslant\hskip 1.99997ptn\hskip 1.99997pt-\hskip 1.99997pt1$.   One  gets $g_{\hskip 0.70004pti\hskip 0.70004pt+\hskip 0.70004pt1}$  by  replacing  in $g_{\hskip 0.70004pti}$  the  two consecutive factors $u\hskip 1.49994pt(\hskip 1.00006ptm_{\hskip 0.70004pti}\hskip 1.49994pt)\hskip 1.00006pt\cdot\hskip 1.00006ptu\hskip 1.49994pt(\hskip 1.00006ptk_{\hskip 0.70004pti}\hskip 1.49994pt)$  by  the factors  $u\hskip 1.49994pt(\hskip 1.00006ptl_{\hskip 0.70004pti}\hskip 1.49994pt)\hskip 1.00006pt\cdot\hskip 1.00006ptu\hskip 1.49994pt(\hskip 1.00006ptm_{\hskip 0.70004pti\hskip 0.70004pt+\hskip 0.70004pt1}\hskip 1.49994pt)$.   Clearly ,   the paths  $m_{\hskip 0.70004pti}\hskip 1.00006pt\cdot\hskip 1.00006ptk_{\hskip 0.70004pti}$  and  $l_{\hskip 0.70004pti}\hskip 1.00006pt\cdot\hskip 1.00006ptm_{\hskip 0.70004pti\hskip 0.70004pt+\hskip 0.70004pt1}$  are short  and  homotopic  by  a  short  homotopy .   It  follows  that

$$\quad u\hskip 1.49994pt(\hskip 1.00006ptm_{\hskip 0.70004pti}\hskip 1.49994pt)\hskip 1.00006pt\cdot\hskip 1.00006ptu\hskip 1.49994pt(\hskip 1.00006ptk_{\hskip 0.70004pti}\hskip 1.49994pt)\hskip 3.99994pt=\hskip 3.99994ptu\hskip 1.49994pt(\hskip 1.00006ptm_{\hskip 0.70004pti}\hskip 1.00006pt\cdot\hskip 1.00006ptk_{\hskip 0.70004pti}\hskip 1.49994pt)\hskip 3.99994pt=\hskip 3.99994ptu\hskip 1.49994pt(\hskip 1.00006ptl_{\hskip 0.70004pti}\hskip 1.00006pt\cdot\hskip 1.00006ptm_{\hskip 0.70004pti\hskip 0.70004pt+\hskip 0.70004pt1}\hskip 1.49994pt)\hskip 3.99994pt=\hskip 3.99994ptu\hskip 1.49994pt(\hskip 1.00006ptl_{\hskip 0.70004pti}\hskip 1.49994pt)\hskip 1.00006pt\cdot\hskip 1.00006ptu\hskip 1.49994pt(\hskip 1.00006ptm_{\hskip 0.70004pti\hskip 0.70004pt+\hskip 0.70004pt1}\hskip 1.49994pt)$$

and  hence  $g_{\hskip 0.70004pti}\hskip 3.99994pt=\hskip 3.99994ptg_{\hskip 0.70004pti\hskip 0.70004pt+\hskip 0.70004pt1}$.   As we saw,   this implies  that  $u^{\hskip 0.70004pt\sharp}\hskip 1.00006pt(\hskip 1.00006ptp_{\hskip 0.35002ptj}\hskip 1.49994pt)\hskip 3.99994pt=\hskip 3.99994ptu^{\hskip 0.70004pt\sharp}\hskip 1.00006pt(\hskip 1.00006ptp_{\hskip 0.35002ptj\hskip 0.70004pt-\hskip 0.70004pt1}\hskip 1.49994pt)$.   In  turn,   this implies  that  $u^{\hskip 0.70004pt\sharp}\hskip 1.00006pt(\hskip 1.00006ptp\hskip 1.49994pt)\hskip 3.99994pt=\hskip 3.99994ptu^{\hskip 0.70004pt\sharp}\hskip 1.00006pt(\hskip 1.00006ptq\hskip 1.49994pt)$.   This completes  the proof  of  the fact  that  $u^{\hskip 0.70004pt\sharp}$  is  a cocycle.

Since  the cocycle  $u^{\hskip 0.70004pt\sharp}$  does not  depend on  the partitions of  $I$  used  in  the construction,   the restriction of  $u^{\hskip 0.70004pt\sharp}$  to  $P_{\hskip 0.70004pt\mathcal{U}}\hskip 1.00006pt(\hskip 1.49994ptX\hskip 1.49994pt)$  is  equal  to  the original  short  cocycle $u$  (for a short  path one can use  the partition of  $I$  into one interval,   namely  $I$).   Conversely ,   suppose  that  $v\hskip 1.99997pt\in\hskip 1.99997ptZ^{\hskip 0.70004pt1}\hskip 1.00006pt(\hskip 1.49994ptX\hskip 0.50003pt,\hskip 3.00003ptY\hskip 1.49994pt)$  and  $u$  is  the restriction of  $v$  to  $P_{\hskip 0.70004pt\mathcal{U}}\hskip 1.00006pt(\hskip 1.49994ptX\hskip 1.49994pt)$.   Since  $v$  is  a  cocycle,   (2)  implies  that  $u^{\hskip 0.70004pt\sharp}\hskip 3.99994pt=\hskip 3.99994ptv$.   It  follows  that  the restriction  map  $Z^{\hskip 0.70004pt1}\hskip 1.00006pt(\hskip 1.49994ptX\hskip 0.50003pt,\hskip 3.00003ptY\hskip 1.49994pt)\hskip 3.99994pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 3.99994ptZ^{\hskip 0.70004pt1}_{\hskip 0.70004pt\mathcal{U}}\hskip 1.00006pt(\hskip 1.49994ptX\hskip 0.50003pt,\hskip 3.00003ptY\hskip 1.49994pt)$  is  a  bijection.

Clearly ,   the restriction of  cocycles  to $P_{\hskip 0.70004pt\mathcal{U}}\hskip 1.00006pt(\hskip 1.49994ptX\hskip 1.49994pt)$  is  equivariant  with  respect  to  the action of  $0$-cochains  (which are all  short )  on  $Z^{\hskip 0.70004pt1}\hskip 1.00006pt(\hskip 1.49994ptX\hskip 0.50003pt,\hskip 3.00003ptY\hskip 1.49994pt)$  and  $Z^{\hskip 0.70004pt1}_{\hskip 0.70004pt\mathcal{U}}\hskip 1.00006pt(\hskip 1.49994ptX\hskip 0.50003pt,\hskip 3.00003ptY\hskip 1.49994pt)$.   Since  the above restriction map  is  a  bijection,   the equivariance implies  that  the induced map  $H^{\hskip 0.70004pt1}\hskip 1.00006pt(\hskip 1.49994ptX\hskip 0.50003pt,\hskip 3.00003ptY\hskip 1.49994pt)\hskip 3.99994pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 3.99994ptH^{\hskip 0.70004pt1}_{\hskip 0.70004pt\mathcal{U}}\hskip 1.00006pt(\hskip 1.49994ptX\hskip 0.50003pt,\hskip 3.00003ptY\hskip 1.49994pt)$  is  also  a  bijection.   This completes  the proof  of  the  theorem.    $\blacksquare$

4. The  standard  Seifert–van Kampen  theorem

The restriction maps.   Suppose  that  $A$  is  a path-connected subspace of  $X$  and  $b\hskip 1.99997pt\in\hskip 1.99997ptA$.   The inclusion map  $i\hskip 1.00006pt\colon\hskip 1.00006ptA\hskip 1.99997pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 1.99997ptX$  induces  the homomorphism  $i_{\hskip 0.70004pt*}\hskip 1.00006pt\colon\hskip 1.00006pt\pi_{\hskip 0.70004pt1}\hskip 1.49994pt(\hskip 1.49994ptA\hskip 0.50003pt,\hskip 1.99997ptb\hskip 1.49994pt)\hskip 1.99997pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 1.99997pt\pi_{\hskip 0.70004pt1}\hskip 1.49994pt(\hskip 1.49994ptX\hskip 0.50003pt,\hskip 1.99997ptb\hskip 1.49994pt)$,   which,   in  turn,   induces  the  restriction  map

$$\quad r_{\hskip 1.39998ptA}\hskip 1.00006pt\colon\hskip 1.00006pt\operatorname{Hom}\hskip 1.49994pt(\hskip 1.49994pt\pi_{\hskip 0.70004pt1}\hskip 1.49994pt(\hskip 1.49994ptX\hskip 0.50003pt,\hskip 1.99997ptb\hskip 1.49994pt)\hskip 0.50003pt,\hskip 3.00003ptG\hskip 1.49994pt)\hskip 3.99994pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 3.99994pt\operatorname{Hom}\hskip 1.49994pt(\hskip 1.49994pt\pi_{\hskip 0.70004pt1}\hskip 1.49994pt(\hskip 1.49994ptA\hskip 0.50003pt,\hskip 1.99997ptb\hskip 1.49994pt)\hskip 0.50003pt,\hskip 3.00003ptG\hskip 1.49994pt)\hskip 3.00003pt.$$

Similarly,  $i$ induces a map  $i_{\hskip 0.70004pt*}\hskip 1.00006pt\colon\hskip 1.00006ptP\hskip 1.00006pt(\hskip 1.49994ptA\hskip 1.49994pt)\hskip 1.99997pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 1.99997ptP\hskip 1.00006pt(\hskip 1.49994ptX\hskip 1.49994pt)$ such  that  $i_{\hskip 0.70004pt*}\hskip 1.00006pt(\hskip 1.00006ptp\hskip 1.00006pt\cdot\hskip 1.00006ptq\hskip 1.49994pt)\hskip 3.99994pt=\hskip 3.99994pti_{\hskip 0.70004pt*}\hskip 1.00006pt(\hskip 1.00006ptp\hskip 1.49994pt)\hskip 1.00006pt\cdot\hskip 1.00006pti_{\hskip 0.70004pt*}\hskip 1.00006pt(\hskip 1.00006ptq\hskip 1.49994pt)$  when  $p\hskip 1.00006pt\cdot\hskip 1.00006ptq$  is  defined,   and $i\hskip 0.50003pt,\hskip 1.99997pti_{\hskip 0.70004pt*}$ induce  the  restriction  maps

$$\quad r_{\hskip 1.39998ptA}\hskip 1.00006pt\colon\hskip 1.00006ptC^{\hskip 0.70004pt0}\hskip 1.00006pt(\hskip 1.49994ptX\hskip 0.50003pt,\hskip 3.00003ptb\hskip 1.49994pt)\hskip 3.99994pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 3.99994ptC^{\hskip 0.70004pt0}\hskip 1.00006pt(\hskip 1.49994ptA\hskip 0.50003pt,\hskip 3.00003ptb\hskip 1.49994pt)\hskip 1.00006pt,\quad r_{\hskip 1.39998ptA}\hskip 1.00006pt\colon\hskip 1.00006ptF^{\hskip 0.70004pt1}\hskip 1.00006pt(\hskip 1.49994ptX\hskip 0.50003pt,\hskip 3.00003ptb\hskip 1.49994pt)\hskip 3.99994pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 3.99994ptF^{\hskip 0.70004pt1}\hskip 1.00006pt(\hskip 1.49994ptA\hskip 0.50003pt,\hskip 3.00003ptb\hskip 1.49994pt)\hskip 3.00003pt$$

respectively,   where  $F$  stands  for  $C$,  $Z$,  or  $H$.   Let  $\mathcal{U}$ be an open covering of  $X$.   Then  $\{\hskip 1.99997ptU\hskip 1.00006pt\cap\hskip 1.00006ptA\hskip 1.00006pt\mid\hskip 1.00006ptU\hskip 1.99997pt\in\hskip 1.99997pt\mathcal{U}\hskip 1.99997pt\}$  is  an open covering of  $A$,   which we will  still  denote by  $\mathcal{U}$.   The inclusion  $i$  induces a map  $i_{\hskip 0.70004pt*}\hskip 1.00006pt\colon\hskip 1.00006ptP_{\hskip 0.70004pt\mathcal{U}}\hskip 1.00006pt(\hskip 1.49994ptA\hskip 1.49994pt)\hskip 1.99997pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 1.99997ptP_{\hskip 0.70004pt\mathcal{U}}\hskip 1.00006pt(\hskip 1.49994ptX\hskip 1.49994pt)$,   which,   in  turn,   induces  restriction  maps

$$\quad r_{\hskip 1.39998ptA}\hskip 1.00006pt\colon\hskip 1.00006ptF^{\hskip 0.70004pt1}_{\hskip 0.70004pt\mathcal{U}}\hskip 1.00006pt(\hskip 1.49994ptX\hskip 0.50003pt,\hskip 3.00003ptb\hskip 1.49994pt)\hskip 3.99994pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 3.99994ptF^{\hskip 0.70004pt1}_{\hskip 0.70004pt\mathcal{U}}\hskip 1.00006pt(\hskip 1.49994ptA\hskip 0.50003pt,\hskip 3.00003ptb\hskip 1.49994pt)\hskip 3.00003pt,$$

where  $F\hskip 3.99994pt=\hskip 3.99994ptC\hskip 0.50003pt,\hskip 3.00003ptZ$  or  $H$.

4.1. Theorem.   Suppose  that  $U\hskip 0.50003pt,\hskip 3.00003ptV\hskip 1.99997pt\subset\hskip 1.99997ptX$  are  two path-connected open sets such  that  $U\hskip 1.99997pt\cap\hskip 1.99997ptV$  is  path-connected and  $b\hskip 1.99997pt\in\hskip 1.99997ptU\hskip 1.99997pt\cap\hskip 1.99997ptV$.   Then  the square of  restriction maps

is  commutative and cartesian.   

Proof.   Let  $\mathcal{U}\hskip 3.99994pt=\hskip 3.99994pt\{\hskip 1.49994ptU\hskip 0.50003pt,\hskip 3.00003ptV\hskip 1.49994pt\}$.   Let  us  replace  the cohomology  set  $H^{\hskip 0.70004pt1}\hskip 1.00006pt(\hskip 1.49994ptX\hskip 0.50003pt,\hskip 3.00003ptb\hskip 1.00006pt)$  by  its  short  version  $H^{\hskip 0.70004pt1}_{\hskip 0.70004pt\mathcal{U}}\hskip 1.00006pt(\hskip 1.49994ptU\hskip 0.50003pt,\hskip 3.00003ptb\hskip 1.00006pt)$.   It  is  sufficient  to prove  that  the resulting square

where all  maps are  the restriction maps,   is  commutative and cartesian.   The commutativity  is  obvious both  for  the original  square and  its short  version.   To simplify  notations,   we will  omit  the base point  $b$  in  the rest  of  the proof .   The  last  square  is  cartesian  if  the map

$$\quad(\hskip 1.00006ptr_{\hskip 1.04996ptU}\hskip 0.50003pt,\hskip 3.00003ptr_{\hskip 1.04996ptV}\hskip 1.49994pt)\hskip 1.00006pt\colon\hskip 1.00006ptH^{\hskip 0.70004pt1}_{\hskip 0.70004pt\mathcal{U}}\hskip 1.00006pt(\hskip 1.49994ptX\hskip 1.49994pt)\hskip 3.99994pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 3.99994ptH^{\hskip 0.70004pt1}\hskip 1.00006pt(\hskip 1.49994ptU\hskip 1.49994pt)\hskip 1.99997pt\times\hskip 1.99997ptH^{\hskip 0.70004pt1}\hskip 1.00006pt(\hskip 1.49994ptV\hskip 1.49994pt)$$

induces a bijection  from  $H^{\hskip 0.70004pt1}_{\hskip 0.70004pt\mathcal{U}}\hskip 1.00006pt(\hskip 1.49994ptX\hskip 1.49994pt)$  to  the fibered  product  $H^{\hskip 0.70004pt1}\hskip 1.00006pt(\hskip 1.49994ptU\hskip 1.49994pt)\hskip 1.99997pt\times_{\hskip 2.10002ptH^{\hskip 0.50003pt1}\hskip 0.70004pt(\hskip 1.04996ptU\hskip 1.39998pt\cap\hskip 1.39998ptV\hskip 1.04996pt)}\hskip 1.99997ptH^{\hskip 0.70004pt1}\hskip 1.00006pt(\hskip 1.49994ptV\hskip 1.49994pt)$.

Surjectivity .   Suppose  that  $u\hskip 1.99997pt\in\hskip 1.99997ptZ^{\hskip 0.70004pt1}\hskip 1.00006pt(\hskip 1.49994ptU\hskip 1.49994pt)$  and  $v\hskip 1.99997pt\in\hskip 1.99997ptZ^{\hskip 0.70004pt1}\hskip 1.00006pt(\hskip 1.49994ptV\hskip 1.49994pt)$  are such  that

$$\quad r_{\hskip 1.04996ptU\hskip 0.70004pt\cap\hskip 0.70004ptV}\hskip 1.00006pt(\hskip 1.00006ptu\hskip 1.49994pt)\quad\ \mbox{and}\quad\ r_{\hskip 1.04996ptU\hskip 0.70004pt\cap\hskip 0.70004ptV}\hskip 1.00006pt(\hskip 1.00006ptv\hskip 1.49994pt)$$

belong  to  the same cohomology  class.   Then  there exists a $0$-cochain  $c\hskip 1.99997pt\in\hskip 1.99997ptC^{\hskip 0.70004pt0}\hskip 1.00006pt(\hskip 1.49994ptU\hskip 1.99997pt\cap\hskip 1.99997ptV\hskip 1.49994pt)$  such  that  $c\hskip 1.00006pt\bullet\hskip 1.00006ptv\hskip 1.00006pt(\hskip 1.00006ptp\hskip 1.49994pt)\hskip 3.99994pt=\hskip 3.99994ptu\hskip 1.00006pt(\hskip 1.00006ptp\hskip 1.49994pt)$  for every  path  $p\hskip 1.99997pt\in\hskip 1.99997ptP\hskip 1.49994pt(\hskip 1.49994ptU\hskip 1.99997pt\cap\hskip 1.99997ptV\hskip 1.49994pt)$.   Let  us extend  the cochain  $c$  to a cochain  $c^{\hskip 0.70004pt\sim}\hskip 1.99997pt\in\hskip 1.99997ptC^{\hskip 0.70004pt0}\hskip 1.00006pt(\hskip 1.49994ptV\hskip 1.49994pt)$.   The cochains  $u$  and  $c^{\hskip 0.70004pt\sim}\hskip 1.00006pt\bullet\hskip 1.00006ptv$  agree on  $P\hskip 1.49994pt(\hskip 1.49994ptU\hskip 1.99997pt\cap\hskip 1.99997ptV\hskip 1.49994pt)\hskip 3.99994pt=\hskip 3.99994ptP\hskip 1.49994pt(\hskip 1.49994ptU\hskip 1.49994pt)\hskip 1.99997pt\cap\hskip 1.99997ptP\hskip 1.49994pt(\hskip 1.49994ptV\hskip 1.49994pt)$  and  hence define a $1$-cochain  $w\hskip 1.99997pt\in\hskip 1.99997ptC^{\hskip 0.70004pt1}_{\hskip 0.70004pt\mathcal{U}}\hskip 1.00006pt(\hskip 1.49994ptX\hskip 1.49994pt)$.   The cochain  $w$  is  a  short  cocycle  because  the conditions for beings a short  cocycle are imposed only  in  $U$  and  in  $V$.   Clearly ,

$$\quad r_{\hskip 1.04996ptU}\hskip 1.00006pt(\hskip 1.00006ptw\hskip 1.49994pt)\hskip 3.99994pt=\hskip 3.99994ptu\quad\ \mbox{and}\quad\ r_{\hskip 1.04996ptV}\hskip 1.00006pt(\hskip 1.00006ptw\hskip 1.49994pt)\hskip 3.99994pt=\hskip 3.99994ptc^{\hskip 0.70004pt\sim}\hskip 1.00006pt\bullet\hskip 1.00006ptv\hskip 3.00003pt.$$

Since  $c^{\hskip 0.70004pt\sim}\hskip 1.00006pt\bullet\hskip 1.00006ptv$  belongs  to  the same cohomology  class as  $v$,   the surjectivity  follows.

Injectivity .   Suppose  that  $w\hskip 0.24994pt,\hskip 3.00003ptz\hskip 3.00003pt\in\hskip 3.00003ptZ^{\hskip 0.70004pt1}_{\hskip 0.70004pt\mathcal{U}}\hskip 1.00006pt(\hskip 1.49994ptX\hskip 1.49994pt)$  are such  that  $r_{\hskip 1.04996ptU}\hskip 1.00006pt(\hskip 1.00006ptw\hskip 1.49994pt)$  and  $r_{\hskip 1.04996ptU}\hskip 1.00006pt(\hskip 1.00006ptz\hskip 1.49994pt)$  belong  to  the same cohomology  class  in  $H^{\hskip 0.70004pt1}\hskip 1.00006pt(\hskip 1.49994ptU\hskip 1.49994pt)$,   and  $r_{\hskip 1.04996ptV}\hskip 1.00006pt(\hskip 1.00006ptw\hskip 1.49994pt)$  and  $r_{\hskip 1.04996ptV}\hskip 1.00006pt(\hskip 1.00006ptz\hskip 1.49994pt)$  belong  to  the same cohomology  class  in  $H^{\hskip 0.70004pt1}\hskip 1.00006pt(\hskip 1.49994ptV\hskip 1.49994pt)$.   Then  there exist  $0$-cochains  $a\hskip 1.99997pt\in\hskip 1.99997ptC^{\hskip 0.70004pt0}\hskip 1.00006pt(\hskip 1.49994ptU\hskip 1.49994pt)$  and  $c\hskip 1.99997pt\in\hskip 1.99997ptC^{\hskip 0.70004pt0}\hskip 1.00006pt(\hskip 1.49994ptV\hskip 1.49994pt)$  such  that

$$\quad w\hskip 1.49994pt(\hskip 1.00006ptp\hskip 1.49994pt)\hskip 3.99994pt=\hskip 3.99994pta\hskip 1.49994pt(\hskip 1.49994ptp\hskip 1.00006pt(\hskip 1.00006pt0\hskip 1.00006pt)\hskip 1.49994pt)\hskip 1.00006pt\cdot\hskip 1.00006ptz\hskip 1.49994pt(\hskip 1.00006ptp\hskip 1.49994pt)\hskip 1.00006pt\cdot\hskip 1.00006pta\hskip 1.49994pt(\hskip 1.49994ptp\hskip 1.00006pt(\hskip 1.00006pt1\hskip 1.00006pt)\hskip 1.49994pt)^{\hskip 0.70004pt-\hskip 0.70004pt1}\quad\ \mbox{and}$$

$$\quad w\hskip 1.49994pt(\hskip 1.00006ptp\hskip 1.49994pt)\hskip 3.99994pt=\hskip 3.99994ptc\hskip 1.49994pt(\hskip 1.49994ptp\hskip 1.00006pt(\hskip 1.00006pt0\hskip 1.00006pt)\hskip 1.49994pt)\hskip 1.00006pt\cdot\hskip 1.00006ptz\hskip 1.49994pt(\hskip 1.00006ptp\hskip 1.49994pt)\hskip 1.00006pt\cdot\hskip 1.00006ptc\hskip 1.49994pt(\hskip 1.49994ptp\hskip 1.00006pt(\hskip 1.00006pt1\hskip 1.00006pt)\hskip 1.49994pt)^{\hskip 0.70004pt-\hskip 0.70004pt1}\hskip 3.00003pt$$

if  $p$  belongs  to  $P\hskip 1.49994pt(\hskip 1.49994ptU\hskip 1.49994pt)$  and  $P\hskip 1.49994pt(\hskip 1.49994ptV\hskip 1.49994pt)$  respectively .   We claim  that  $a\hskip 1.00006pt(\hskip 1.00006ptx\hskip 1.49994pt)\hskip 3.99994pt=\hskip 3.99994ptc\hskip 1.49994pt(\hskip 1.00006ptx\hskip 1.49994pt)$  for every  $x\hskip 1.99997pt\in\hskip 1.99997ptU\hskip 1.99997pt\cap\hskip 1.99997ptV$.   Indeed,   since  $U\hskip 1.99997pt\cap\hskip 1.99997ptV$  is  path-connected,   there exists  $p\hskip 1.99997pt\in\hskip 1.99997ptP\hskip 1.49994pt(\hskip 1.49994ptU\hskip 1.99997pt\cap\hskip 1.99997ptV\hskip 1.49994pt)$  such  that  $p\hskip 1.00006pt(\hskip 1.00006pt0\hskip 1.00006pt)\hskip 3.99994pt=\hskip 3.99994ptb$  and  $p\hskip 1.00006pt(\hskip 1.00006pt1\hskip 1.00006pt)\hskip 3.99994pt=\hskip 3.99994ptx$.   Then  $a\hskip 1.49994pt(\hskip 1.49994ptp\hskip 1.00006pt(\hskip 1.00006pt0\hskip 1.00006pt)\hskip 1.49994pt)\hskip 3.99994pt=\hskip 3.99994ptc\hskip 1.49994pt(\hskip 1.49994ptp\hskip 1.00006pt(\hskip 1.00006pt0\hskip 1.00006pt)\hskip 1.49994pt)\hskip 3.99994pt=\hskip 3.99994pt1$.   It  follows  that

$$\quad z\hskip 1.49994pt(\hskip 1.00006ptp\hskip 1.49994pt)\hskip 1.00006pt\cdot\hskip 1.00006pta\hskip 1.49994pt(\hskip 1.49994ptx\hskip 1.49994pt)^{\hskip 0.70004pt-\hskip 0.70004pt1}\hskip 3.99994pt=\hskip 3.99994ptz\hskip 1.49994pt(\hskip 1.00006ptp\hskip 1.49994pt)\hskip 1.00006pt\cdot\hskip 1.00006ptc\hskip 1.49994pt(\hskip 1.49994ptx\hskip 1.49994pt)^{\hskip 0.70004pt-\hskip 0.70004pt1}$$

and  hence  $a\hskip 1.49994pt(\hskip 1.49994ptx\hskip 1.49994pt)\hskip 3.99994pt=\hskip 3.99994ptc\hskip 1.49994pt(\hskip 1.49994ptx\hskip 1.49994pt)$  for every  $x\hskip 1.99997pt\in\hskip 1.99997ptU\hskip 1.99997pt\cap\hskip 1.99997ptV$.   In other  terms,  $a$  and  $c$  agree on  the intersection of  their domains and  hence define a $0$-cochain  $d\hskip 1.99997pt\in\hskip 1.99997ptC^{\hskip 0.70004pt0}\hskip 1.00006pt(\hskip 1.49994ptX\hskip 0.50003pt,\hskip 3.00003ptb\hskip 1.00006pt)$  such  that

$$\quad w\hskip 1.49994pt(\hskip 1.00006ptp\hskip 1.49994pt)\hskip 3.99994pt=\hskip 3.99994ptd\hskip 1.49994pt(\hskip 1.49994ptp\hskip 1.00006pt(\hskip 1.00006pt0\hskip 1.00006pt)\hskip 1.49994pt)\hskip 1.00006pt\cdot\hskip 1.00006ptz\hskip 1.49994pt(\hskip 1.00006ptp\hskip 1.49994pt)\hskip 1.00006pt\cdot\hskip 1.00006ptd\hskip 1.49994pt(\hskip 1.49994ptp\hskip 1.00006pt(\hskip 1.00006pt1\hskip 1.00006pt)\hskip 1.49994pt)^{\hskip 0.70004pt-\hskip 0.70004pt1}$$