Talk:Supplement I (Ex)

Before stating the exercise, we recall the definition of $\Sigma^m$$\newcommand{\Z}{\mathbb{Z}} \newcommand{\R}{\mathbb{R}} \newcommand{\C}{\mathbb{C}} \newcommand{\N}{\mathbb{N}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\F}{\mathbb{F}} \newcommand{\bZ}{\mathbb{Z}} \newcommand{\bR}{\mathbb{R}} \newcommand{\bC}{\mathbb{C}} \newcommand{\bH}{\mathbb{H}} \newcommand{\bQ}{\mathbb{Q}} \newcommand{\bF}{\mathbb{F}} \newcommand{\bN}{\mathbb{N}} \DeclareMathOperator\id{id} % identity map \DeclareMathOperator\Sq{Sq} % Steenrod squares \DeclareMathOperator\Homeo{Homeo} % group of homeomorphisms of a topoloical space \DeclareMathOperator\Diff{Diff} % group of diffeomorphisms of a smooth manifold \DeclareMathOperator\SDiff{SDiff} % diffeomorphism under some constraint \DeclareMathOperator\Hom{Hom} % homomrphism group \DeclareMathOperator\End{End} % endomorphism group \DeclareMathOperator\Aut{Aut} % automorphism group \DeclareMathOperator\Inn{Inn} % inner automorphisms \DeclareMathOperator\Out{Out} % outer automorphism group \DeclareMathOperator\vol{vol} % volume \newcommand{\GL}{\text{GL}} % general linear group \newcommand{\SL}{\text{SL}} % special linear group \newcommand{\SO}{\text{SO}} % special orthogonal group \newcommand{\O}{\text{O}} % orthogonal group \newcommand{\SU}{\text{SU}} % special unitary group \newcommand{\Spin}{\text{Spin}} % Spin group \newcommand{\RP}{\Rr\mathrm P} % real projective space \newcommand{\CP}{\Cc\mathrm P} % complex projective space \newcommand{\HP}{\Hh\mathrm P} % quaternionic projective space \newcommand{\Top}{\mathrm{Top}} % topological category \newcommand{\PL}{\mathrm{PL}} % piecewise linear category \newcommand{\Cat}{\mathrm{Cat}} % any category \newcommand{\KS}{\text{KS}} % Kirby-Siebenmann class \newcommand{\Hud}{\text{Hud}} % Hudson torus \newcommand{\Ker}{\text{Ker}} % Kernel \newcommand{\underbar}{\underline} %Classifying Spaces for Families of Subgroups \newcommand{\textup}{\text} \newcommand{\sp}{^}\Sigma^m$, and explain why it is indeed a simplicial complex.

Definition 0.1. The simplicial complex $\Sigma^m$$\Sigma^m$ is defined to have $k$$k$-simplices equal to the set of $(m-k)$$(m-k)$-simplices of $\partial \Delta^{m+1}$$\partial \Delta^{m+1}$, denoted by $\sigma^*$$\sigma^*$ for $\sigma\in\partial\Delta^{m+1}$$\sigma\in\partial\Delta^{m+1}$. The face relations are given by $\sigma^* \leq \tau^*$$\sigma^* \leq \tau^*$ iff $\sigma \geq \tau$$\sigma \geq \tau$.

Lemma 0.2. This is a simplicial complex.

Proof. The following is an equivalent definition of $\Sigma^m$$\Sigma^m$:

• Its vertices are the $m$$m$-dimensional faces in $\partial\Delta^{m+1}$$\partial\Delta^{m+1}$.
• A collection of vertices of $\Sigma^m$$\Sigma^m$ -- i.e. $m$$m$-dimensional faces in $\partial\Delta^{m+1}$$\partial\Delta^{m+1}$ -- is a simplex of $\Sigma^m$$\Sigma^m$ iff the intersecion of all the faces is non-trivial.

This is an equivalent definition because of the following special ('miraculous'!) property of $\partial\Delta^{m+1}$$\partial\Delta^{m+1}$:

• The $(m-k)$$(m-k)$-simplices are precisely the $k$$k$-fold intersections of $m$$m$-simplices.

This equivalent description makes it clear that $\Sigma^m$$\Sigma^m$ is a simplicial complex, since a subcollection of a non-trivially intersecting collection is again non-trivially intersecting.

$\square$$\square$

Remark 0.3. The first description of $\Sigma^m$$\Sigma^m$ gives us a 'map' of simplicial complexes

$\displaystyle \tau \mapsto \tau^* \;\colon\; \partial\Delta^{m+1} \longrightarrow \Sigma^m .$ $$\displaystyle \tau \mapsto \tau^* \;\colon\; \partial\Delta^{m+1} \longrightarrow \Sigma^m .$$

This is of course not a real map of simplicial complexes, since it sends $k$$k$-simplices to $(m-k)$$(m-k)$-simplices. However, it does induce a map of barycentric subdivisions

$\displaystyle (\tau_0 < \tau_1 < \cdots < \tau_k) \mapsto (\tau_0^* > \tau_1^* > \cdots > \tau_k^*) \;\colon\; (\partial\Delta^{m+1})^\prime \longrightarrow (\Sigma^m)^\prime ,$ $$\displaystyle (\tau_0 < \tau_1 < \cdots < \tau_k) \mapsto (\tau_0^* > \tau_1^* > \cdots > \tau_k^*) \;\colon\; (\partial\Delta^{m+1})^\prime \longrightarrow (\Sigma^m)^\prime ,$$

which is an isomorphism of simplicial complexes. So we may view subcomplexes of $(\partial\Delta^{m+1})^\prime$$(\partial\Delta^{m+1})^\prime$ as living in $(\Sigma^m)^\prime$$(\Sigma^m)^\prime$ via this isomorphism.

Exercise 0.4. Let $K$$K$ be a subcomplex of $\partial\Delta^{m+1}$$\partial\Delta^{m+1}$, and consider the barycentric subdivision $K^\prime$$K^\prime$ as a subcomplex of $(\Sigma^m)^\prime$$(\Sigma^m)^\prime$. Show that

$\displaystyle D(\sigma ,K) = \sigma^* \cap K^\prime .$ $$\displaystyle D(\sigma ,K) = \sigma^* \cap K^\prime .$$

Solution 0.5. First consider what happens if we view everything as living inside $(\partial\Delta^{m+1})^\prime$$(\partial\Delta^{m+1})^\prime$:

• The barycentric subdivision $K^\prime$$K^\prime$ is the subcomplex with $k$$k$-simplices $\lbrace \tau_0 < \tau_1 < \cdots < \tau_k \rbrace$$\lbrace \tau_0 < \tau_1 < \cdots < \tau_k \rbrace$ with $\tau_i$$\tau_i$ a simplex of $K$$K$ for each $i$$i$.
• The dual complex $D(\sigma ,K)$$D(\sigma ,K)$ is by definition the subcomplex of $K^\prime$$K^\prime$ whose $k$$k$-simplices are $\lbrace \tau_0 < \tau_1 < \cdots < \tau_k \rbrace$$\lbrace \tau_0 < \tau_1 < \cdots < \tau_k \rbrace$ with $\tau_i$$\tau_i$ a simplex of $K$$K$ and $\tau_i \geq \sigma$$\tau_i \geq \sigma$ for each $i$$i$.
• Finally, $\sigma$$\sigma$ is the subcomplex with $k$$k$-simplices $\lbrace \tau_0 < \tau_1 < \cdots < \tau_k \rbrace$$\lbrace \tau_0 < \tau_1 < \cdots < \tau_k \rbrace$ with $\tau_i \leq \sigma$$\tau_i \leq \sigma$ for all $i$$i$.

So we clearly have $D(\sigma ,K) \neq \sigma\cap K^\prime \subseteq (\partial\Delta^{m+1})^\prime$$D(\sigma ,K) \neq \sigma\cap K^\prime \subseteq (\partial\Delta^{m+1})^\prime$, since we have different conditions $\tau_i \geq \sigma$$\tau_i \geq \sigma$ and $\tau_i \leq \sigma$$\tau_i \leq \sigma$ on the LHS and RHS respectively. This is solved by passing from $(\partial\Delta^{m+1})^\prime$$(\partial\Delta^{m+1})^\prime$ to $(\Sigma^m)^\prime$$(\Sigma^m)^\prime$ and replacing $\sigma$$\sigma$ by $\sigma^*$$\sigma^*$. Considered as subcomplexes of $(\Sigma^m)^\prime$$(\Sigma^m)^\prime$, we have:

• $K^\prime$$K^\prime$ has $k$$k$-simplices $\lbrace \tau_0^* > \tau_1^* > \cdots > \tau_k^* \rbrace$$\lbrace \tau_0^* > \tau_1^* > \cdots > \tau_k^* \rbrace$ with $\tau_i$$\tau_i$ a simplex of $K$$K$ for each $i$$i$.
• $D(\sigma ,K)$$D(\sigma ,K)$ has $k$$k$-simplices $\lbrace \tau_0^* > \tau_1^* > \cdots > \tau_k^* \rbrace$$\lbrace \tau_0^* > \tau_1^* > \cdots > \tau_k^* \rbrace$ with $\tau_i$$\tau_i$ a simplex of $K$$K$ and $\tau_i \geq \sigma$$\tau_i \geq \sigma$, equivalently $\tau_i^* \leq \sigma^*$$\tau_i^* \leq \sigma^*$, for all $i$$i$.
• $\sigma^*$$\sigma^*$ has $k$$k$-simplices $\lbrace \tau_0^* > \tau_1^* > \cdots > \tau_k^* \rbrace$$\lbrace \tau_0^* > \tau_1^* > \cdots > \tau_k^* \rbrace$ with $\tau_i^* \leq \sigma^*$$\tau_i^* \leq \sigma^*$ for all $i$$i$.

Hence,

$\displaystyle D(\sigma ,K) = \sigma^* \cap K^\prime \subseteq (\Sigma^m)^\prime .$ $$\displaystyle D(\sigma ,K) = \sigma^* \cap K^\prime \subseteq (\Sigma^m)^\prime .$$

Remark 0.6. Once we know that the dual cell decomposition of $\partial\Delta^{m+1}$$\partial\Delta^{m+1}$ is actually a simplicial complex, in other words that $\Sigma^m$$\Sigma^m$ is a simplicial complex, then the argument is very simple and formal. The main geometric input to the argument is the special property of $\partial\Delta^{m+1}$$\partial\Delta^{m+1}$ mentioned above, that $(m-k)$$(m-k)$-simplices are precisely the $k$$k$-fold intersections of $m$$m$-simplices.