Immersibility of manifolds is decidable in odd codimension

Fix open covers $\mathcal{U}$ and $\mathcal{V}$ of $M$ and $N$ admitting orthonormal frames.

Let $[\phi]$ be a homotopy class of tangent bundle monomorphism containing a specific monomorphism $\phi$. Then pick some $\phi\in[\phi]$ and we will construct a lift $\psi:M\rightarrow\operatorname{Mono}(TM,TN)$. Then for each $u\in\mathcal{U}$ and each $v\in\mathcal{V}$ with $f(u)\cap v$ nonempty $\phi$ provides, for each $p\in u$ with $f(p)\in v$, an $m$-frame $O$ over $(p,f(p))$ viewed as a point in the local trivialization over $u\times v$, and again using a retraction to $O(n)$ we can make the frame orthonormal. If we had some $u^{\prime}$ such that $p\in u^{\prime}$ and $v^{\prime}$ such that $f(p)\in v^{\prime}$ then we would produce a frame $O^{\prime}$, and these would be related precisely by $O^{\prime}=\psi^{*}O\varphi$ where $\varphi$ is the transition map from $u$ to $u^{\prime}$ for the tangent bundle on $M$ and $\psi$ is the transition map from $v$ to $v^{\prime}$ for the tangent bundle on $N$. This is exactly the transition map from $u\times v$ to $u^{\prime}\times v^{\prime}$ for $\operatorname{Mono}(TM,TN)$ on $M\times N$, so this produces a lift. If we had picked a different $\phi^{\prime}\in[\phi]$ they would be related by a homotopy, and again reducing to coordinates over $\mathcal{U}\times\mathcal{V}$, we could produce a homotopy of lifts.

Conversely, given a homotopy class $[g]$ of lift of the above diagram, we pick some lift $g$ in the class and produce a tangent bundle monomorphism. Again, for each $p$ in $M$, each $u\in U$ containing $p$ and each $v\in\mathcal{V}$ containing $f(p)$ the lift $g$ provides an $m$-frame over $(p,f(p))$ which we can view as a map from $T_{p}M$ to $T_{f(p)}N$ in the coordinates of $u$ and $v$. Again that this is a coherent choice of frame across all choices of $u,v$ follows from the fact that the transition maps agree. Finally, a different $g^{\prime}\in[g]$ would be related by a homotopy to $g$, and this can again be written out in coordinates. ∎

Again we pick a covering collection of opens $\mathcal{U}$ on $BSO(m)$ and $\mathcal{V}$ on $BSO(n)$ admitting local trivializations, and consider the open cover $\mathcal{W}$ of $BSO(m)\times BSO(n)$ by taking sets of the form $W=U\times V$. Then over each of these sets we have a trivial $V(m,n)$ bundle and we construct transition maps from $U\times V$ to $U^{\prime}\times V^{\prime}$ as before: if $\varphi$ is the transition map from $U$ to $U^{\prime}$ and $\psi$ is the transition map from $V$ to $V^{\prime}$ then over each point in $(U\times V)\cap(U^{\prime}\times V^{\prime})=(U\times U^{\prime})\cap(V\times V^{\prime})$ we have the transition map $\zeta(O)=\psi^{*}O\varphi$.

The map $\operatorname{Mono}(TM,TN)\rightarrow\operatorname{Mono}(m-\operatorname{planes},n-\operatorname{planes})$ works as follows: we can pull back the collection $\mathcal{U}$ on $BSO(m)$ across $\kappa_{m}$ to form a collection of opens on $M$, and the local trivializations of $ESO(m)\twoheadrightarrow BSO(m)$ pull back to trivializations of the tangent bundle of $M$. Similarly, the collection of opens $\mathcal{V}$ pulls back across $\kappa_{n}$ along with the trivializations of $ESO(n)\twoheadrightarrow BSO(n)$. Then for any point $(p,q)\in M\times N$ we can pick some $\kappa_{m}^{-1}(u)\times\kappa_{n}^{-1}(v)$ containing $(p,q)$ and we can use this choice of trivialization to map the fiber over $(p,q)$ to the fiber over $\kappa_{m}(p)\times\kappa_{n}(q)$. That this describes a map coherently over all choices of trivialization of $BSO(m)$ and $BSO(n)$ follows from the fact that the tangent bundles of $M$ and $N$ are pullbacks of $ESO(m)\twoheadrightarrow BSO(m)$ and $ESO(n)\twoheadrightarrow BSO(n)$ respectively.

Finally, this is a pullback square because firstly it would factor through the pullback, and the map to the pullback would be a map of fiber bundles with the same base, and hence a projection of fibers, and since the fiber is the same it is hence a homeomorphism. ∎

This is simply a matter of solving a matrix equation. Indeed, with $V$ as given in the statement of the lemma, we construct a pair of maps $D,T:V\rightarrow\oplus_{i=1}^{k}A$ as follows: on $v=(v_{1},\ldots,v_{k})$ define

and set

so that the system of equations is equivalent to

which we can now turn into a matrix equation. In particular, as each $A^{(n_{i})}$ can be represented as a rational vector space, as can $A$ itself, and since the differential is $\mathbb{Q}$-linear, so is $D$, as is $T$ by construction. Then picking a basis for $V$ and $A$ as $\mathbb{Q}$-vector spaces, we have the above equation as a matrix equation, and the space of solutions can be constructed via row reduction. ∎

Fix a basis for $W$, written as a choice of elements $w^{(i)}_{j}$ where $i$ ranges over the positive dimensional degrees, and $j$ ranges from $1$ to the dimension of $W^{(i)}$. Constructing a map for the dashed line as in the above diagram is simply a matter of picking a target for each element $w_{j}^{(i)}$, as $\mathcal{M}_{X}\otimes\wedge W$ is the free commutative graded $\mathcal{M}_{X}$-algebra. The only conditions then to check on such a map $\psi$ are $d_{\mathcal{M}_{X}}\circ\psi=\psi\circ d$ and $f^{*}=i^{*}\circ\psi$. Then for each $w^{(i)}_{j}$ we have the equations $d_{\mathcal{M}_{X}}(\psi(w_{j}^{(i)}))=\psi(d(w^{(i)}_{j}))$ and $f^{*}(w^{(i)}_{j})=i^{*}(\psi(w^{(i)}_{j}))$.

By assumption, $d$ is linear through the cohomological dimension of $(X,A)$, and in higher dimensions there is a unique lift of any element in $\mathcal{M}_{A}$. Then we have a fixed $\psi(w_{j}^{(i)})$ for $i>n$.

Otherwise we know that $dw^{(i)}_{j}$ takes the form

where each $m^{i}_{j}$ and $m^{i,k,l}_{j}\in\mathcal{M}_{X}$. Putting all of this together then, we obtain the set of equations:

and by lemma 4.1 we can construct the affine space of solutions to these equations as a subspace of $V=\oplus_{i,j}A^{(i)}$ (note that we have a copy of $A^{(i)}$ for each generator in $W^{(i)}$.) We can call this space $S$.

We consider the affine space $\tilde{V}\subset V$ given by:

and we simply have to compute $S\cap\tilde{V}$, which is the intersection of two affine subspaces of a $\mathbb{Q}$-vector space, which can be computed.

∎

We begin by observing that since $m$ is defined on elements of the fibrewise product $(a,b)\in Y\times_{B}Y$, we have that $p(a)=p(b)$ so that we have $(\mathrm{id}\times(e\circ p)\times\mathrm{id})\circ(\mathrm{id}\times\Delta)=(\mathrm{id}\times(e\circ p)\times\mathrm{id})\circ(\Delta\times\mathrm{id})$. Then checking associativity of multiplication is a straightforward calculation. Indeed we have:

simply by substitution, and then applying the commutativity of operations on different copies of the product. Since the fibrewise Mal’cev operation has to also be homotopy associative, this is homotopic to

Applying the equality above we have that this is equal to

and finally this can be rewritten as

and again applying the above equality we have this is equal to

To check that the section acts as an identity, we first observe that $((e\circ p)\times\mathrm{id})\circ\Delta\circ(e\circ p)=\Delta\circ(e\circ p)=(\mathrm{id}\times(e\circ p))\circ\Delta\circ(e\circ p)$ since $(e\circ p)\circ(e\circ p)=(e\circ p)$ and for any map $f$ we have $(f\times f)\circ\Delta=\Delta\circ f$.

Now we compute

and by the axioms of a fibrewise $HM$-space, $\tau\circ(\mathrm{id}\times\Delta)$ is fibrewise homotopic to $\pi_{1}$ and the above map is fibrewise homotopic to

An analogous argument shows that $m\circ((e\circ p)\times\mathrm{id})\circ\Delta$ is also fibrewise homotopic to the identity, and hence $p$ with section $e$ and multiplication $m$ is a fibrewise $H$-space.

∎

We will proceed in two stages. We start by showing the rationalization $p_{\mathbb{Q}}:Y_{\mathbb{Q}}\twoheadrightarrow B_{\mathbb{Q}}$ is a fibrewise HM-space, and then we lift that structure to the fibrewise rationalization. To see that the rationalization is a fibrewise HM-space is straightforward: starting with the relative minimal model, we can construct a coMal’cev operation. In particular we have the relative minimal model presented as $M_{B}\otimes\wedge W_{F}$ where $W_{F}$ is the graded vector space built out of the rational homotopy groups of $F$. The linearity condition on the differential ensures that the map $M_{B}\otimes\wedge W_{F}\rightarrow(M_{B}\otimes\wedge W_{F})^{\otimes_{\mathcal{M}_{B}}3}$ induced by the map sending $w\in W$ to $w\otimes 1\otimes 1-1\otimes w\otimes 1+1\otimes 1\otimes w$ is a map of dgas. This map satisfies properties dual to those of the Mal’cev operation, and so it gives us a fibrewise heap structure on the rationalization. We want to first show that the k-invariants on each Postnikov stage of the rationalization are $HM$-maps. The spaces $K(\pi_{n}(F)\otimes\mathbb{Q},n+1)$ are H-spaces, and so the operation $(a,b,c)=a-b+c$ endows it with the structure of an $HM$-space. This gives the trivial $B_{\mathbb{Q}}$ fibration $B_{\mathbb{Q}}\times K(\pi_{n}(F)\otimes\mathbb{Q},n+1)$ the structure of a relative $HM$-space. A simple computation shows that the coMal’cev operation this induces on the relative minimal model on this fibration is precisely the map sending $\alpha$ to $\alpha\otimes 1\otimes 1-1\otimes\alpha\otimes 1+1\otimes 1\otimes\alpha$, from which it follows that the $k$-invariant on the corresponding stage of the Moore-Postnikov tower is an $HM$-map.

Because the maps on the bottom and right of this pullback square are $HM$-maps, so is the map $L_{n,\mathbb{Q}}\rightarrow L_{n-1,\mathbb{Q}}$. Them we need to show that this structure pulls back appropriately to the fibrewise rationalization. In particular since the fibrewise rationalization is a pullback, we can define the Mal’cev operation on $L_{n}^{\mathbb{Q}}$ by pulling back the operation on the rationalization:

The associativity and Mal’cev properties are satisfied trivially since this is a pullback of a fibrewise $HM$-space structure on the rationalization.

∎

Note that in the language of the Serre spectral sequence we can rewrite this condition as saying that we have an element in $\alpha\in\bigoplus_{p+q=n}E^{p,q}_{\infty}$ such that $t\alpha\in E_{\infty}^{n,0}$ for some $t$. Then it suffices to show the following: suppose we have some torsion element of $\beta\in E^{p,q}_{\infty}$ where $q\geq 1$, then there exists some $r$ such that $\chi_{r}^{*}\alpha=0$. Indeed, since the direct sum above is finite, then we can take the direct sum decomposition of the element $\alpha$ into a finite sum of terms $\beta_{q}$ in $E_{p,q}^{\infty}$, for which some $r_{q}$ will suffice, and then $\chi_{r_{1}\dots r_{n}}$ will kill all but $\beta_{0}$ as desired. Recall that the $E_{1}$ page of the Serre spectral sequence has terms $E_{1}^{p,q}=C^{p}(B;H^{q}(F;A))$.

Suppose then that we have some torsion cohomology class $\beta$ as above, there is some corresponding cocycle $\gamma$ which survives to $E^{p,q}_{\infty}$, but $t\gamma$ does not survive. By lemma $3.5$ of [6] we know that there is some $s$ such that $\chi_{s}^{*}(H^{q}(F;A))\subseteq tH^{q}(F;A)$, from which we can conclude that $\chi_{s}^{*}(C^{p}(B;H^{q}(F;A)))\subseteq tC^{p}(B;H^{q}(F;A))$, and hence $\chi_{s}^{*}(\gamma)$ does not survive to $E^{p,q}_{\infty}$, as desired.

∎

Let $d$ be the dimension of $X$. To simplify the proof a bit, we pullback $p$ across $f$ to obtain a fibration $\hat{p}:\hat{Y}\rightarrow X$, for which we will determine if a section exists.

We will denote by $P_{n}$ the $n^{th}$ Moore-Postnikov stage of $\hat{p}:\hat{Y}\twoheadrightarrow X$. The linear relative minimal model pulls back to model $\hat{p}$ so we still have a relative minimal model for this fibration which is linear through the dimension of $X$. Using this minimal model, we can apply lemma 4.2 to construct a map of dgas dual to a lift. Here the algorithm might fail to find a lift, in which case we know none exists. Indeed, if a lift existed, applying the equivalence would produce a dual map on dgas, so that if no such map exists, no section can exist.

We assume then that we found such a map. In particular we have $\epsilon:\mathcal{M}_{X}\otimes\wedge W_{F}\rightarrow\mathcal{M}_{X}$ which is a map of dgas, where $\mathcal{M}_{X}\otimes\wedge W_{F}$ with differential $d_{L}^{*}$ is the relative minimal model of $\hat{p}$.

We will attempt to construct a section for $\hat{p}$ inductively as follows: for each $n$ through dimension $d$ we will construct a fibration $h_{n}:L_{n}\twoheadrightarrow X$, as well as a section $e_{n}$ and rational equivalences $\phi_{n}:L_{n}\rightarrow P_{n}$, $\theta_{n}:P_{n}\rightarrow L_{n}$ over $X$, where the composition $\theta_{n}\circ\phi_{n}$ is multiplication by some integer $s_{n}$ under the fibrewise $H$-space structure on $L_{n}$ with the section $e_{n}$. In particular, these will be constructed so that each $h_{n}$ is a fibrewise $HM$-space with operation $\tau_{n}$, and there are maps $r_{n}:L_{n}\rightarrow L_{n-1}$ which commute with the fibrewise $HM$-structure and form a Moore-Postnikov tower for $h_{d}$. At each stage we will also fix a fibrewise rationalization $u_{n}$ to the corresponding Moore-Postnikov stage $L_{n}^{\mathbb{Q}}$ which is isomorphic to $P_{n}^{\mathbb{Q}}$. By lemma 4.7 each $L_{n}^{\mathbb{Q}}$ has the structure of a fibrewise $HM$-space, and we will ensure that the maps $u_{n}$ are each $HM$-maps.

At each stage, in the construction of $\phi_{n}$, it is possible for the construction to fail, and this indicates that no section of $\hat{p}$ exists. We will see why when we construct $\phi_{n}$.

Since the space is simply connected, we set $h_{1}:X\twoheadrightarrow X$ to the identity and the other data is trivial.

Then we assume we have constructed $h_{n-1}$, $e_{n-1}$, $\phi_{n-1}$,$\theta_{n-1}$ and $u_{n-1}$. We start by constructing the map $h_{n}$. We do this by picking $L_{n}\twoheadrightarrow L_{n-1}$ as a $K(\pi_{n}(F),n)$ fibration. This will then serve as the top stage on the Moore-Postnikov tower for $h_{n}:L_{n}\twoheadrightarrow X$. This is equivalent to picking a classifying map in $[L_{n-1},K(\pi_{n}(F),n+1)]_{B}$, or alternatively, a cohomology class in $H^{n+1}(L_{n-1};\pi_{n}(F))$. By the universal coefficient theorem this is isomorphic to