Talk:Structured chain complexes IV (Ex)

Recall the suspension maps

$\displaystyle S : \Sigma W(C) \to W(\Sigma C)$ $$\displaystyle S : \Sigma W(C) \to W(\Sigma C)$$

where $W$$\newcommand{\Z}{\mathbb{Z}} \newcommand{\R}{\mathbb{R}} \newcommand{\C}{\mathbb{C}} \newcommand{\N}{\mathbb{N}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\F}{\mathbb{F}} \newcommand{\bZ}{\mathbb{Z}} \newcommand{\bR}{\mathbb{R}} \newcommand{\bC}{\mathbb{C}} \newcommand{\bH}{\mathbb{H}} \newcommand{\bQ}{\mathbb{Q}} \newcommand{\bF}{\mathbb{F}} \newcommand{\bN}{\mathbb{N}} \DeclareMathOperator\id{id} % identity map \DeclareMathOperator\Sq{Sq} % Steenrod squares \DeclareMathOperator\Homeo{Homeo} % group of homeomorphisms of a topoloical space \DeclareMathOperator\Diff{Diff} % group of diffeomorphisms of a smooth manifold \DeclareMathOperator\SDiff{SDiff} % diffeomorphism under some constraint \DeclareMathOperator\Hom{Hom} % homomrphism group \DeclareMathOperator\End{End} % endomorphism group \DeclareMathOperator\Aut{Aut} % automorphism group \DeclareMathOperator\Inn{Inn} % inner automorphisms \DeclareMathOperator\Out{Out} % outer automorphism group \DeclareMathOperator\vol{vol} % volume \newcommand{\GL}{\text{GL}} % general linear group \newcommand{\SL}{\text{SL}} % special linear group \newcommand{\SO}{\text{SO}} % special orthogonal group \newcommand{\O}{\text{O}} % orthogonal group \newcommand{\SU}{\text{SU}} % special unitary group \newcommand{\Spin}{\text{Spin}} % Spin group \newcommand{\RP}{\Rr\mathrm P} % real projective space \newcommand{\CP}{\Cc\mathrm P} % complex projective space \newcommand{\HP}{\Hh\mathrm P} % quaternionic projective space \newcommand{\Top}{\mathrm{Top}} % topological category \newcommand{\PL}{\mathrm{PL}} % piecewise linear category \newcommand{\Cat}{\mathrm{Cat}} % any category \newcommand{\KS}{\text{KS}} % Kirby-Siebenmann class \newcommand{\Hud}{\text{Hud}} % Hudson torus \newcommand{\Ker}{\text{Ker}} % Kernel \newcommand{\underbar}{\underline} %Classifying Spaces for Families of Subgroups \newcommand{\textup}{\text} \newcommand{\sp}{^}W$ can mean $W_{\%}$$W_{\%}$, $W^{\%}$$W^{\%}$, or $\widehat{W}^{\%}$$\widehat{W}^{\%}$ defined in TM L-Theory II. We wish to compute

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$$\displaystyle \underset{k}{\rm hocolim} \;\; \Sigma^{-k} \widehat{W}^{\%} (\Sigma^kC) .$$

Consider the exact sequence

$\displaystyle \xymatrix{ 0 \ar[r] & W^{\%}(C) \ar[r] & \widehat{W}^{\%}(C) \ar[r] & \Sigma W_{\%} (C) \ar[r] & 0 }$

By functoriality, we get maps $W(C) \to \Sigma^{-1} W(\Sigma^1 C)$$W(C) \to \Sigma^{-1} W(\Sigma^1 C)$ for all different brands of $W$$W$, and these are the maps that represent the above colimit. These maps induce a diagram

$\displaystyle \xymatrix{ 0 \ar[r] & W^{\%}(C) \ar[d] \ar[r] & \widehat{W}^{\%} \ar[r] \ar[d] & \Sigma W_{\%}(C) \ar[d] \ar[r] & 0 \\ 0 \ar[r] & \Sigma^{-k} W^{\%}(\Sigma^k C) \ar[r] & \Sigma^{-k}\widehat{W}^{\%}(\Sigma^k C) \ar[r] & \Sigma^{-k+1} W_{\%}(\Sigma^k C) \ar[r] & 0 . \\ }$

We have seen that the suspension map

$\displaystyle S : \Sigma \widehat{W}^{\%} (C) \to \widehat{W}^{\%}(\Sigma C)$ $$\displaystyle S : \Sigma \widehat{W}^{\%} (C) \to \widehat{W}^{\%}(\Sigma C)$$

is a homotopy equivalence. Iterating this we see hocolim $\Sigma^{-k} \widehat{W}^{\%}(\Sigma^k C) = \widehat{W}^{\%}(C)$$\Sigma^{-k} \widehat{W}^{\%}(\Sigma^k C) = \widehat{W}^{\%}(C)$.

Now, note that for chain complexes $\Sigma(A \otimes B) = (\Sigma A) \otimes B = A \otimes (\Sigma B)$$\Sigma(A \otimes B) = (\Sigma A) \otimes B = A \otimes (\Sigma B)$. Thus

$\displaystyle \Sigma^{-k} W_{\%}(\Sigma^k C) \simeq \Sigma^k(W \otimes_{\mathbb{Z}[\mathbb{Z}/2]} (C \otimes C))$ $$\displaystyle \Sigma^{-k} W_{\%}(\Sigma^k C) \simeq \Sigma^k(W \otimes_{\mathbb{Z}[\mathbb{Z}/2]} (C \otimes C))$$

This complex increases linearly in connectivity as $k$$k$ increases, hence its homotopy colimit is contractible. That is

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$$\displaystyle {\rm hocolim} \; \Sigma^{-k} W_{\%}(\Sigma^k C) \simeq 0 .$$

By exactness of the short exact sequences in our diagram, and the remarks above, we conclude that

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$$\displaystyle \widehat{W}^{\%}(C) \simeq \underset{k}{\rm hocolim} \;\; \Sigma^{-k} W^{\%} (\Sigma^kC) .$$