# Lecture 1 (Tuesday ?)

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

See [Homological Stability Course Notes](Homological%20Stability%20Course%20Notes.md)

References:

- [Course web page](https://www.utsc.utoronto.ca/people/kupers/seminars/minicourse-on-homological-stability/) 

# Notes

:::{.remark}
Realize the symmetric group $\Sigma_n$ as the group of bijections of $\ul{n} \da \ts{1,\cdots, n}$ under composition.
For any group $G$, a model of $\BG$ is $\realize{\nerve{\pt \htyquot G}}$, and we can ask what $H_*(\BG; \ZZ)$ is.
:::

:::{.example title="?"}
Identifying $\B C_2 \cong \RP^\infty$, we have
\[
H_*(\B C_2; \ZZ) = 
\begin{cases}
\ZZ & *=0 
\\
C_2 & *>0 \text{ odd }
\\
0 & \text{else}.
\end{cases}
.\]
The only slightly more complicated group $D_3$ has more complicated homology, supported in infinitely many degrees.
:::

:::{.remark}
Looking at a table of $H_d(\Sigma_n; \ZZ)$ 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 
\[
\sigma^n: H_*(\B\Sigma_n; \ZZ) \to H_*(\B\Sigma_{n+1}; \ZZ)
.\]
Several conjectures become apparent: the $\sigma^*$ are isomorphisms in some range depending on $n$, are injective, and are isomorphisms on $p^k\dash$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:
:::

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

:::{.theorem title="Homological stability of symmetric groups"}
The sequence $\ts{\B \Sigma_n}_{n\in \ZZ_{\geq 0}}$ exhibits homological stability, and in fact the maps $\sigma^n$ are surjective in degrees $d\leq n/2$ and isomorphisms in degrees $d\leq {n-1\over 2}$.
:::

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

![](Projects/9999IP%20Homological%20Stability/Homological%20Stability%20Lectures/figures/2022-01-10_21-59-26.png)

:::

:::{.remark}
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
\[
\B \Sigma \da \Disjoint_{n\geq 0} \B \Sigma_n
,\]
which is an $E_1$ space (a (unital) topological monoid).
It is a fact that $\pi_0 \B \Sigma \iso \NN$ as commutative monoids, and the reason it is commutative "comes from" the fact that $\B \Sigma$ was homotopy-commutative.
Stabilization further induces a right-shift map $\B\Sigma \to \B\Sigma[1]$, which in turn makes $H_*(\B\Sigma; \ZZ)$ a $\ZZ[\pi_0]\dash$module for $\pi_0\da \pi_0 \B\Sigma$ and can be described as multiplication by an element $\sigma\in \B\Sigma_1$.
This becomes invertible in the limit, yielding an isomorphism 
\[
H^*(\B\Sigma; \ZZ)\invert{\pi_0 \B\Sigma}\iso \colim_{\sigma} H_*(\B\Sigma; \ZZ)
.\]
:::

:::{.theorem title="McDuff-Segal"}
Suppose $M$ is a (unital) topological monoid that is associative and homotopy commutative, and let $\Loop$ denote the based loop space construction.
Regard $\pt \htyquot M$ as a category with one object and set $\B M \da \realize{\nerve{\pt \htyquot M}}$ be the bar construction, then
\[
H_*(M; \ZZ)\invert{ {\pi_0} } \cong H^*(\Loop \B M; \ZZ)
.\]
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 $\B M$.
:::

:::{.theorem title="Barratt-Priddy-Quillen-Segal"}
There is a homotopy equivalence
\[
\Loop \B\qty{\B\Sigma} \homotopic \Loop^\infty \SS
.\]
Alternatively, $\SS = \K_\alg(\Fin\Set)$.
:::