# Lecture 1 (Tuesday ?)

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, $${\mathbb{S}}= {\mathsf{K}}_\mathsf{Alg}({\mathsf{Fin}}{\mathsf{Set}})$$.

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