Tags: #todo #projects/active #homotopy/homological-stability

See __Homological Stability Course Notes__

References:

# Notes

Realize the symmetric group \(\Sigma_n\) as the group of bijections of \(\underline{n} \coloneqq\left\{{1,\cdots, n}\right\}\) under composition. For any group \(G\), a model of \({{\mathbf{B}}G}\) is \({ {\left\lvert {{ \mathcal{N}({{\operatorname{pt}}{ \mathbin{/\mkern-6mu/}}G}) }} \right\rvert} }\), and we can ask what \(H_*({{\mathbf{B}}G}; {\mathbb{Z}})\) is.

Identifying \({\mathbf{B}}C_2 \cong {\mathbb{RP}}^\infty\), we have \begin{align*} H_*({\mathbf{B}}C_2; {\mathbb{Z}}) = \begin{cases} {\mathbb{Z}}& *=0 \\ C_2 & *>0 \text{ odd } \\ 0 & \text{else}. \end{cases} .\end{align*} The only slightly more complicated group \(D_3\) has more complicated homology, supported in infinitely many degrees.

Looking at a table of \(H_d(\Sigma_n; {\mathbb{Z}})\) suggests stabilization as \(d\) is fixed and \(n\to\infty\). To make this precise, use the inclusions \(\Sigma_n \to \Sigma_{n+1}\) that extend a permutation by the identity to induce group morphisms \begin{align*} \sigma^n: H_*({\mathbf{B}}\Sigma_n; {\mathbb{Z}}) \to H_*({\mathbf{B}}\Sigma_{n+1}; {\mathbb{Z}}) .\end{align*} Several conjectures become apparent: the \(\sigma^*\) are isomorphisms in some range depending on \(n\), are injective, and are isomorphisms on \(p^k{\hbox{-}}\)torsion unless \(p\divides n+1\). Note that the injectivity is a special property that we might not expect in general, and is related to the existence of transfers. This motivates the following definition:

A sequence of spaces \(\cdots\to X_{n} \to X_{n+1} \to \cdots\) exhibits **homological stability** iff the induced maps \(\sigma^k: H_*(X_n; {\mathbb{Z}})\to H_*(X_{n+1}; {\mathbb{Z}})\) are isomorphisms in a range of degrees depending on \(n\).

The sequence \(\left\{{{\mathbf{B}}\Sigma_n}\right\}_{n\in {\mathbb{Z}}_{\geq 0}}\) exhibits homological stability, and in fact the maps \(\sigma^n\) are surjective in degrees \(d\leq n/2\) and isomorphisms in degrees \(d\leq {n-1\over 2}\).

A consequence is that for large \(n\), the map \(H_*({\mathbf{B}}\Sigma_n; {\mathbb{Z}}) \to \colim_n H_*({\mathbf{B}}\Sigma_{n}; {\mathbb{Z}})\) is an isomorphism. An example of using this result: the sign homomorphisms \(\operatorname{sgn}: \Sigma_n\to C_2\) induces map on homology, and we one can use this to prove \([\Sigma_n, \Sigma_n] = A_n\). Use that \(A_n = \ker \operatorname{sgn}\) on one hand, and show \(\operatorname{sgn}: \Sigma_n\to C_2\) coincides with abelianization for \(n\geq 2\) and so its kernel consists of commutators. To do this, use that \(\pi_1 {{\mathbf{B}}G}= G\) that the Hurewicz map \(\pi_1 X\to H_1(X;{\mathbb{Z}})\) is abelianization. This induces a map \(G^{{\operatorname{ab}}} { \, \xrightarrow{\sim}\, }H_1(X; {\mathbb{Z}})\), so it suffices to show that \(\operatorname{sgn}\) induces an isomorphism on homology. This follows from a diagram chase on the following ladder:

Using that there is a system of maps \(\Sigma_n\to \Sigma_{n+k}\) and disjoint unions of sets induce group morphisms \(\Sigma_n \times \Sigma_m \to \Sigma_{n+m}\), one can define \begin{align*} {\mathbf{B}}\Sigma \coloneqq\displaystyle\coprod_{n\geq 0} {\mathbf{B}}\Sigma_n ,\end{align*} which is an \(E_1\) space (a (unital) topological monoid). It is a fact that \(\pi_0 {\mathbf{B}}\Sigma { \, \xrightarrow{\sim}\, }{\mathbb{N}}\) as commutative monoids, and the reason it is commutative “comes from” the fact that \({\mathbf{B}}\Sigma\) was homotopy-commutative. Stabilization further induces a right-shift map \({\mathbf{B}}\Sigma \to {\mathbf{B}}\Sigma[1]\), which in turn makes \(H_*({\mathbf{B}}\Sigma; {\mathbb{Z}})\) a \({\mathbb{Z}}[\pi_0]{\hbox{-}}\)module for \(\pi_0\coloneqq\pi_0 {\mathbf{B}}\Sigma\) and can be described as multiplication by an element \(\sigma\in {\mathbf{B}}\Sigma_1\). This becomes invertible in the limit, yielding an isomorphism \begin{align*} H^*({\mathbf{B}}\Sigma; {\mathbb{Z}}){ \left[ { \scriptstyle \frac{1}{\pi_0 {\mathbf{B}}\Sigma} } \right] } { \, \xrightarrow{\sim}\, }\colim_{\sigma} H_*({\mathbf{B}}\Sigma; {\mathbb{Z}}) .\end{align*}

Suppose \(M\) is a (unital) topological monoid that is associative and homotopy commutative, and let \({\Omega}\) denote the based loop space construction. Regard \({\operatorname{pt}}{ \mathbin{/\mkern-6mu/}}M\) as a category with one object and set \({\mathbf{B}}M \coloneqq{ {\left\lvert {{ \mathcal{N}({{\operatorname{pt}}{ \mathbin{/\mkern-6mu/}}M}) }} \right\rvert} }\) be the bar construction, then \begin{align*} H_*(M; {\mathbb{Z}}){ \left[ { \scriptstyle \frac{1}{ {\pi_0} } } \right] } \cong H^*({\Omega}{\mathbf{B}}M; {\mathbb{Z}}) .\end{align*} Part of why this is interesting: there are tools (e.g. infinite loop space machines) to compute the homotopy types of spaces of the form \({\mathbf{B}}M\).

There is a homotopy equivalence \begin{align*} {\Omega}{\mathbf{B}}\qty{{\mathbf{B}}\Sigma} \simeq{\Omega}^\infty {\mathbb{S}} .\end{align*} Alternatively, \({\mathbb{S}}= {\mathsf{K}}_\mathsf{Alg}({\mathsf{Fin}}{\mathsf{Set}})\).