Algorithms in $A_{\infty}$-algebras

In his 1980 paper [18], Kadeishvili proved that the homology of any dg-algebra has an induced $A_{\infty}$-algebra structure. The proof itself is by induction on the arity of the operation, writing out explicitly how to create $m_{n}(a_{1},\dots,a_{n})$ in terms of lower order operations. By giving these expressions, the proof is basically a formulation of what an algorithm for the computation of an $A_{\infty}$ algebra structure on the homology of a dg-algebra would look like.

Even though this initial setup has a strongly algorithmic flavor, actual algorithms and computer-supported calculations of $A_{\infty}$-algebra structures emerged far later. In this paper, we plan to give an overview of work on creating software computing $A_{\infty}$-algebra structures and the contexts and techniques used for these. Merkulov [24] has given Kadeishvili’s construction a more concrete, and combinatorially accessible presentation – but the focus for this paper is in explicit algorithmic computation with computer implementations, preferably publically accessible.

We fix a ring $\mathbb{k}$. All tensor products are over $\mathbb{k}$ unless otherwise noted, and tensor powers are denoted by $A^{\otimes n}=A\otimes\dots\otimes A$.

A graded $k$-vector space $A$ is an $A_{\infty}$-algebra if one of the following equivalent conditions hold

1. 1.

   There is a family of maps $\mu_{i}\colon A^{\otimes i}\to A$, called higher multiplications fulfilling the Stasheff identities

   $$\operatorname{St}_{n}\colon\sum_{i}\sum_{j}\mu_{i}\circ_{j}\mu_{n-i}=0\quad.$$

2. 2.

   There is a family of chain maps from the cellular chain complex of the associahedra to appropriate higher endomorphisms of $A$

   $$\mu_{n}\colon C_{*}(K_{n})\to\operatorname{Hom}(A^{\otimes n},A)\quad.$$

3. 3.

   $A$ is a representation of the free dg-operad resolution $\mathcal{A}ss_{\infty}$ of the associative operad.

An $A_{\infty}$-coalgebra is defined by dualizing this definition.

The structure was introduced by Jim Stasheff in [27]. Good introductory surveys have been written by Lu, Palmieri, Wu and Zhang [22, 21] as well as by Bernhard Keller [20, 19].

A graded $k$-vector space $A$ is a differential graded algebra (dg-algebra) if it is equipped with a differential operator $\partial:A\to A$ of degree $-1$ and an associative multiplication $m_{2}:A\otimes A\to A$ of degree $0$, such that the Leibniz rule holds:

$$\partial m_{2}(x,y)=m_{2}(\partial x,y)+(-1)^{|x|}m_{2}(x,\partial y)$$

A module over a dg-algebra $A$ is a graded vectorspace $M$ with a differential operator $\partial:M\to M$ and an associative multiplication $m_{2}:M\otimes A\to M$ that obeys the Leibniz rule.

Kadeishvili proved in his 1980 paper [18] that the homology of a dg-algebra has an inherited and quasi-isomorphic $A_{\infty}$-algebra structure. The proof starts out defining $\mu_{1}=0$ and $\mu_{2}([x_{1}]\otimes[x_{2}])=(-1)^{|x_{1}|+1}[x_{1}\cdot[x_{2}]$, as well as starting to define an $A_{\infty}$-morphism $f$ by $f_{1}$ simply a cycle-choosing homomorphism.

The proof proceeds by induction¹¹1which translates well to recursion: this is how algorithms enter the picture: if all $m_{j}$ and $f_{j}$ have been defined for $j<i$, then let

$$U_{n}=\sum_{s=1}^{n}m_{2}\circ f_{s}\otimes f_{n-s})+\sum_{k=0}^{n-2}\sum_{j=2}^{n-1}f_{n-j+1}\circ\mathbb{1}^{\otimes k}\otimes m_{j}\otimes\mathbb{1}^{\otimes n-j-k+1}$$

The Stasheff axioms can be translated to

$$m_{1}\circ f_{n}=(f_{1}\circ m_{n}-U_{n})$$

The right hand side is homological to zero, so we can pick $f_{n}$ to be a bounding element of this difference. Extend by linearity.

The structure of this argument lends itself excellently well to concrete and algorithmic calculations, and there has been a few approaches to algorithmic and computer-aided $A_{\infty}$-algebra work. There are three main topics that emerge, and we will dedicated a chapter to each of them.

First up, in Section 3, we will review Ainhoa Berciano’s work on contractions of dg-algebras to dg-modules, with implementations in the computer algebra system Kenzo.

Next, in Section 4 we will go to the realm of group cohomology. Mikael Vejdemo-Johansson worked on algorithms to directly calculate $A_{\infty}$-algebra structures on the modular group cohomology of $p$-groups, and generated a number of ways to recognize feasibility of the calculation as well as a stopping criterion. Stephan Schmid, later on, used some of Vejdemo-Johansson’s results in a concrete calculation of the $A_{\infty}$-algebra structure on the modular group cohomology of the symmetric group $S_{p}$.

Finally, in Section 5, we will describe recent work by Murillo and Belchí on using $A_{\infty}$-coalgebra structures on persistent homology rings to create new perspectives on bottleneck distances and stability of persistence barcodes.

Ainhoa Berciano’s work [3, 8, 4, 6, 5] starts with perturbation theory. This framework has as its core result the Basic Perturbation Lemma [10], that describes how a contraction changes under perturbation.

A contraction connects two dg-modules $M$ and $N$, abstracting the homotopy concepts of deformation retracts. A contraction consists of morphisms $f:N\to M$, $g:M\to N$ and $\phi:N\to N$ such that $f$ and $g$ are almost an isomorphism – up to a homotopy operation in $N$. In other words, we require

$$fg=\mathbb{1}_{M}\quad gf+\phi\partial_{N}+\partial_{N}\phi=\mathbb{1}_{N}\quad f\phi=0\quad\phi g=0\quad\phi\phi=0$$

A contraction preserves homology: $H(N)$ is canonically isomorphic to $H(M)$, but this isomorphism tends not to transfer algebraic structures from $N$ to $M$.

If the differential structure on $N$ is perturbed: instead of boundary operator $\partial_{N}$, $N$ is equipped with a new boundary operator $\partial_{N}+\delta$, then the Basic Perturbation Lemma produces a new contraction. The requirement for this construction is that $\phi\delta$ is pointwise nilpotent: for any $x\in N$ there is an $n$ so that $(\phi\delta)^{n}(x)=0$.

Then there is a new contraction $f_{\delta},g_{\delta},\phi_{\delta}$ between $N$ equipped with the boundary operator $\partial_{N}+\delta$ and $M$ equipped with the boundary operator $\partial_{M}+\partial_{\delta}$ given by

	$\displaystyle\partial_{\delta}$	$\displaystyle=f\delta\sum_{i\geq 0}(-1)^{i}(\phi\delta)^{i}g$	$\displaystyle f_{\delta}$	$\displaystyle=f\left(1-\delta\sum_{i\geq 0}(-1)^{i}(\phi\delta)^{i}\phi\right)$
	$\displaystyle g_{\delta}$	$\displaystyle=\sum_{i\geq 0}(-1)^{i}(\phi\delta)^{i}g$	$\displaystyle\phi_{\delta}$	$\displaystyle=\sum_{i\geq 0}(-1)^{i}(\phi\delta)^{i}\phi$

Berciano generates an algorithm for transferring $A_{\infty}$-coalgebra structures between DG-modules using the tensor trick [16]. The tensor trick starts with a dg-coalgebra $C$, a dg-module $M$ and a contraction from $C$ to $M$. Take the tensor module of the desuspension of all components in this contraction to produce a new contraction. With the cosimplicial differential, we can use the Basic Perturbation Lemma and obtain a new contraction. The tilde cobar differential generates an induced $A_{\infty}$-coalgebra structure, explicitly given by the comultiplication operations

$$\Delta_{i}=(-1)^{[i/2]+i+1}f^{\otimes i}\Delta^{[i]}\phi^{[\otimes(i-1)]}\dots\phi^{[\otimes 2]}\Delta^{[2]}g;\qquad\Delta^{[k]}=\sum_{i=0}^{k-2}(-1)^{i}\mathbb{1}^{\otimes i}\otimes\Delta\otimes\mathbb{1}^{k-i-2}$$

(1)

This derivation allows Berciano to prove [6] that in $H_{*}(K(\pi,n);\mathbb{Z}_{p})$ for a finitely generated abelian group $\pi$, the only non-null morphisms in the $A_{\infty}$-coalgebra structure have to have order $i(p-2)+2$ for some non-negative integer $i$.

The final formula in 1 is concrete enough that it has been implemented on the computer algebra platform Kenzo in the packages ARAIA (Algebra Reduction A-Infinity Algebra) and CRAIC (Coalgebra Reduction A-Infinity Coalgebra).

Fix a group $G$ and a field $\mathbb{k}$. The cohomology algebra of the Eilenberg-MacLane space $H^{*}(K(G,1))$ is isomorphic to the Ext-algebra $\operatorname{Ext}_{\mathbb{k}G}(\mathbb{k},\mathbb{k})$ of the group ring and is called the group cohomology $H^{*}(G)$. Because of the connection to the Ext-algebra, the group cohomology can be calculated from the composition dg-algebra of $\operatorname{Hom}(F_{*},F_{*})$ for a free resolution $F_{*}\to\mathbb{k}$ in the category of $G$-modules. Several computer algebra systems, including Magma [9] and GAP [14] support calculations with $G$-modules. In such a system, we create a free resolution $F_{*}$ of $\mathbb{k}$. Chain maps $F_{*}\to F_{*}$ are then represented by a sequence of maps, one for each degree, each determined by lower-dimensional maps through commutativity of the corresponding squares in the chain map diagram. With $\operatorname{Hom}(F_{*},F_{*})$ represented, we can compute $H^{*}G$ as $H_{*}\operatorname{Hom}(F_{*},F_{*})$.

Since the $\operatorname{Hom}(F_{*},F_{*})$ is a dg-algebra, by Kadeishvili’s theorem, $H^{*}G$ has an induced $A_{\infty}$-algebra structure.

Vejdemo-Johansson [28, 29, 30] studies this $A_{\infty}$-algebra structure from a strictly algorithmic perspective.

Persistent homology and cohomology form the cornerstone of the fast growing field of Topological Data Analysis. The fundamental idea is to study the homology functor applied to diagrams of topological spaces

$$\mathbb{V}:\mathbb{V}_{0}\hookrightarrow\mathbb{V}_{1}\hookrightarrow\dots$$

These spaces are often generated directly from datasets, by constructions such as the Čech construction: for data points $\mathbb{X}=\{x_{0},\dots,x_{N}\}$, an abstract simplical complex $\check{C}_{\epsilon}$ has as its vertices $\mathbb{X}$ and includes a simplex $[x_{i_{0}},\dots,x_{i_{d}}]$ precisely if the intersection of balls $\bigcap_{j=0}^{d}B_{\epsilon}(x_{i_{j}})$ is non-empty. If $\epsilon$ increases, no intersections will become empty, and so no simplices will vanish. So the Čech complexes, as $\epsilon$ sweeps from 0 to $\infty$, generates a nested sequence of topological spaces.

The inclusion maps $\iota_{i}^{j}:\mathbb{V}_{i}\to\mathbb{V}_{j}$ lift by functoriality to linear maps on homology: $H_{*}(\iota_{i}^{j}):H_{*}\mathbb{V}_{i}\to H_{*}\mathbb{V}_{j}$. We may define a persistent homology group as the image $PH_{*}^{i,j}(\mathbb{V})=\operatorname{img}H(\iota_{i}^{j})$.

For more details on the data analysis side, we recommend the surveys [11, 15, 32]