Tags: #active_projects #homological_stability Refs: Homological Stability Course Notes.

Setup and Strategy

References: see Randal-Williams and Wahl.

We’ll follow a strategy going back to Quillen, reconstructed from some sun-bleached physical notes.

The following sequence exhibits homological stability : \begin{align*} {\mathbf{B}}\Sigma_0 \xrightarrow{\sigma} {\mathbf{B}}\Sigma_1 \xrightarrow{\sigma} {\mathbf{B}}\Sigma_2 \to \cdots .\end{align*}

More precisely, the following map is a surjection for \(i\leq n/2\) and an isomorphism if \(i\leq (n-1)/2\): \begin{align*} \sigma_*: H_i ({\mathbf{B}}\Sigma_n; {\mathbb{Z}}) \to H_i({\mathbf{B}}\Sigma_{n+1}; {\mathbb{Z}}) .\end{align*}

Modern: identify \({\mathsf{sSet}}\cong {\mathsf{Spaces}}\) and categories with their nerves. For \(G{\hbox{-}}\)sets, we can take homotopy quotients by the action of \(G\). Can construct \({\mathbf{B}}G\) as geometric realization of a nerve, or a homotopy quotient: \({\mathbf{B}}G \simeq{\operatorname{pt}}{/}G\).

Some properties of \(X {/}G\):

They are natural, in the sense that if \(X \xrightarrow{f} Y\) is \(G{\hbox{-}}\)equivariant, so \(gf(x) = f(gx)\), then there is a map \(X {/}\to Y{/}G\)?
They preserve homological connectivity, where \(f:X\to Y\) is homologically \(n{\hbox{-}}\)connected if \(f_*\) is surjective for \(i\leq d\) and an isomorphism for \(i\leq d-1\). So if \(X \xrightarrow{f} Y\) is \(d{\hbox{-}}\)connected, then \(f{/}G: X{/}G \to Y{/}G\) is \(d{\hbox{-}}\)connected
They commute with geometric realizations : If \(X_\bullet\) is a semi-simplicial \(G{\hbox{-}}\)space, then \({\left\lVert {X_\bullet} \right\rVert} {/}G \simeq{\left\lVert { X_\bullet {/}G} \right\rVert}\).
Homotopy quotients of transitive \(G{\hbox{-}}\)sets, so \begin{align*} {\mathbf{B}}{\operatorname{Stab}}_G(s) \cong \left\{{ s }\right\} {/}{\operatorname{Stab}}_G(s) \simeq S {/}G .\end{align*}

We’ll use these properties to make inductive arguments.

Injective Words

We’ll define \(\Delta\) as non-empty finite ordered sets and order-preserving maps, and write \([p] \coloneqq(0 < \cdots < p)\). A simplicial category is a functor \begin{align*} \Delta^{\operatorname{op}}\xrightarrow{X_\bullet} \mathsf{C} ,\end{align*} where you can take \(\mathsf{C} = {\mathsf{Set}}, {\mathsf{Top}}\), etc. We’ll define \(X_p \coloneqq X([p])\). There are face maps \(\delta_i: [p-1] \to [p]\) which skips \(i\), and degeneracy maps \(\sigma_i: [p] \to [p-1]\) which double the \(i\)th entry. We can then define \begin{align*} {\left\lVert {X_\bullet} \right\rVert} \coloneqq\qty{ {\textstyle\coprod}_{p\geq 0} \Delta^p \times X_p} / \sim .\end{align*} Life will be easier if we forget the degeneracy maps, these will be semi-simplicial sets. Slightly better homotopically behave, can ignore some cofibrancy conditions, and importantly these are smaller than simplicial sets.

First step: in \({\operatorname{pt}}{/}\Sigma_n\), replace \({\operatorname{pt}}\) with \({\left\lVert {{X}_{*}} \right\rVert}\) for \({X}_{*} \in {\mathsf{sSet}}^{\mathrm{semi}}\). Notationally, we’ll write \(X\) as \({ W_n( \underline{1} ) }_{*}\). We’ll build a category \({\mathsf{FI}}\leq {\mathsf{FinSet}}\) in the next lecture, whose objects are finite sets and morphisms are injections. We’ll define \(W_n(\underline{1})_p\) as the semisimplicial set whose \(p{\hbox{-}}\)simplices are given by \(\mathop{\mathrm{Hom}}_{{\mathsf{FI}}}([p], \underline{n} )\) where \(\underline{n} = \left\{{ 1, 2, \cdots, n }\right\}\). The face maps are induced by precomposing with \(\delta_i\).

Set \(W_n(\underline{1}) _p\) to be sequences \((m_0, m_1, \cdots, m_p)\) with each \(m_i \in \underline{n}\).

\(W_2(1)_\bullet\): note that \(W_n\) has up to \(n{\hbox{-}}\)simplices in general, and here

0-simplices: \((1), (2)\),
1-simplices: \((1,2), (2, 1)\).

Taking the geometric realization yields a circle:

\({ W_3(\underline{1})}_{*}\):

0-simplices: \((1), (2), (3)\)
1-simplices: there are 6
2-simplices: again 6

In general, the number of \((n-1){\hbox{-}}\)simplices is \(n!\).

One can check homological connectivity. After the gluing loops will become nullhomotopic:

Here it’s easy to just compute the fundamental groupoid.

\envlist

\({\left\lVert { { W_n(\underline{1}) }_{*}} \right\rVert}\) is homologically \((n-1)/2{\hbox{-}}\)connective.
\(W_n(\underline{1})_p\) is a transitive \(\Sigma_n{\hbox{-}}\)set and the stabilizer of \(x\in W_n(\underline{1})_p\) is isomorphic to \(\Sigma_{n-p-1}\).

Upshot: \({\left\lVert { { W_n(\underline{1}) }_{*}} \right\rVert}\) can serve as a replacement for \({\operatorname{pt}}\) in \({\operatorname{pt}}{/}\Sigma_n\) in the ranges we are interested in.

Next step: exploit the fact that geometric realizations come with a natural filtration, and thus there is a spectral sequence. The filtration is given by \begin{align*} F_r {\left\lVert { {X}_{*}} \right\rVert} \coloneqq\qty{ {\textstyle\coprod}_{0\leq p\leq r} \Delta^p \times X_p } / \sim .\end{align*} We can recover \({{\left\lVert {{X}_{*}} \right\rVert}} = \colim F_i {{\left\lVert {{X}_{*}} \right\rVert}}\):

Link to Diagram

Here \(Y_+\) means adding a disjoint basepoint.

The spectral sequence of a filtration will converge to the reduced homology of the associated graded. (?)

There is a strongly convergent first quadrant spectral sequence: \begin{align*} E_{p, 1}^1 = H_q(X_p; {\mathbb{Z}}) \Rightarrow H_{p+1}( {{\left\lVert {{X}_{*}} \right\rVert}}; {\mathbb{Z}}) .\end{align*}

The differentials are of bidegree \((-r, r-1)\):

There is an explicit formula for some differentials, e.g. \(d^1 = \sum_{i=0}^p (-1)^i (d_i)_*\) where the \(d_i\) are the face maps. The edge homomorphism is induced by \(X_0 \hookrightarrow{{\left\lVert {{X}_{*}} \right\rVert}}\).

Proof of Nakaoka’s Theorem

The proof is by strong induction on \(n\), where we’ll assume \(n+1 \geq 3\).

Step 1

The map \({{\left\lVert {{W_{n+1}(\underline{1})}_{*}} \right\rVert}} \to {\operatorname{pt}}\) is homologically \(({n\over 2} + 1){\hbox{-}}\)connected, thus so is the induced map: \begin{align*} {{\left\lVert {{W_{n+1}(\underline{1}) }_{*}} \right\rVert}} {/}\Sigma_{n+1} \to {\operatorname{pt}}{/}\Sigma_{n+1} .\end{align*}

Step 2

Look at the \(E^1\) page of the geometric realization spectral sequence. Use that geometric realization commutes with homotopy quotients, and we’ll take the spectral sequence for the right-hand side : \begin{align*} {{\left\lVert {{ W_{n+1}(\underline{1})}_{*}} \right\rVert}} {/}\Sigma_{n+1} \cong {\left\lVert { { w_{n+1}(\underline{1}) }_{*} {/}\Sigma_{n+1}} \right\rVert} .\end{align*}

We get \begin{align*} E_{p, 1}^1 = H_1 ( W_{n+1}(\underline{1}) _p {/}\Sigma_{n+1} ) \Rightarrow H_{p+q} \qty{ {\left\lVert { { W_{n+1} (\underline{1})}_{*} {/}\Sigma_{n+1} } \right\rVert}} .\end{align*}

Note that \(W_{n+1}(\underline{1})_p\) is a transitive \(\Sigma_{n+1}{\hbox{-}}\)set. Choose some map \(f: [p] \hookrightarrow\underline{n+1}\) to induce a homotopy equivalence \begin{align*} {\operatorname{pt}}{/}{\operatorname{Stab}}_{\Sigma_{n+1}}(?) \xrightarrow{\sim} W_{n+1}(\underline{1})_p {/}\Sigma_{n+1} .\end{align*} We’ll need to keep track of this choice of \(f\), so set \(f \coloneqq\iota_p\) to be the inclusion of the last \(p+1\) elements \((n-p+1, \cdots, n+1)\). We get \(E_{p, q}^1 = H_q( {\mathbf{B}}\Sigma_{n-p})\), and there is a commutative diagram:

Here the vertical left-hand side map \({\mathrm{inc}}\) is induced by inclusion of stabilizers. A problem is that \(d_p \iota_p \neq \iota_{p-1}\) in general. Instead we have \(h_i d_i \iota_p = \iota_{p-1}\) and we can get a diagram of the following form:

On the left, one should conjugate by \(h_i\) (which is what \(ch_i\) means here) and on the left one should act and conjugate simultaneously. Pick \(h_i\) to be the transposition \((n-p+1, n-p+i)\), then this choice will send \({\operatorname{Stab}}\iota_p\to {\operatorname{Stab}}\iota_{p-1}\) and yields \((d_i)_* = (ch_i \circ {\mathrm{inc}})_*\). The advantage of this choice is that it acts as the identity on \(\Sigma_{n-p}\). We then get

` \begin{align} d^1 = \sum_{0\leq i \leq p} (-1)^i \sigma_

\begin{cases} \sigma_* & p>0 \text { even} \ 0 & \text{otherwise}. \end{cases} \end{align*} `{=html}.

The spectral sequence ends up looking like the following:

The inductive hypothesis will let us say something about \(E^2\).

Next time:

Finish the spectral sequence argument,
Show \({{\left\lVert {{W_n(\underline{1})}_{*}} \right\rVert}}\) is \({n-1 \over 2}\) connected,
Formalize argument for symmetric monoidal groupoids.

Note: homology of configuration spaces over Euclidean space is completely known??