# Eilenberg–Mac Lane Spaces for Topological Groups

## Abstract

**:**

## 1. Introduction

#### 1.1. Background

#### 1.2. Main Results

**Theorem**

**1**

**.**If C is an open cover of $X\in \mathcal{U}$, write ${C}^{\prime}$ for the poset of finite intersections of sets in C, ordered by inclusion. Then $\underline{Sing}(X)$ is weakly equivalent (in the regular structure on $s\mathcal{U}$) to the homotopy colimit (in the compact Hausdorff structure on $s\mathcal{U}$) of ${\left\{\underline{Sing}(U)\right\}}_{U\in {C}^{\prime}}$.

**Theorem**

**2**

**.**Given subspaces $A\subseteq B\subseteq X$ in $\mathcal{U}$ with A closed and B open, the inclusion $(X\setminus A,B\setminus A)\to (X,B)$ induces isomorphisms of the homology group objects ${H}_{n}(X\setminus A,B\setminus A)\to {H}_{n}(X,B)$ for all n. Equivalently, for open subspaces $A,B\subseteq X$ covering X, the inclusion $(B,A\cap B)\to (X,A)$ induces isomorphisms of homology group objects ${H}_{n}(B,A\cap B)\to {H}_{n}(X,A)$ for all n.

**Theorem**

**3**

**.**For open subspaces $A,B\subseteq X$ covering X there is a long exact sequence of homology group objects

**Theorem**

**4**

**.**Suppose $X\in s\mathcal{U}$ with ${X}_{n}$ totally path-disconnected for all n. Then $\underline{Sing}\circ L\left|X\right|$ is weakly equivalent to X in the regular structure.

**Theorem**

**5**

**.**If G is totally path-disconnected, $L\left|\overline{W}G\right|$ is an Eilenberg–Mac Lane space $K(G,1)$ for G.

## 2. Topological Groups and Modules

- (i)
- in any pull-back squareif f is the cokernel of some map then so is ${f}^{\prime}$;
- (ii)
- in any push-out squareif f is the kernel of some map then so is ${f}^{\prime}$.

## 3. Topological Homotopy Groups

## 4. Model Structures

**Theorem**

**6.**

**Proof.**

**Remark**

**1.**

**Theorem**

**7.**

**Proof.**

**Lemma**

**1.**

**Proof.**

**Proposition**

**1.**

## 5. $\underline{Sing}$ and $|-|$

**Theorem**

**8.**

**Remark**

**2.**

**Lemma**

**2.**

**Proof.**

**Lemma**

**3.**

**Proof.**

**Lemma**

**4.**

**Proof.**

**Example**

**1.**

**Proposition**

**2.**

**Proposition**

**3.**

**Proof.**

**Proof of Proposition**

**2.**

- (i)
- a fat n-cell in ${Z}^{m}\setminus {Z}^{m-1}$ with all its faces in ${Z}^{m-1}$;
- (ii)
- two copies of the same fat n-cell in ${Z}^{m}\setminus {Z}^{m-1}$ stuck together at one face in ${Z}^{m}\setminus {Z}^{m-1}$.

- (i)
- For each fat n-cell D in case (i), we have a fat $(n+1)$-cell ${D}^{\prime}$ that has D as one face and all other faces in ${Z}^{m-1}$. Write ${\Sigma}^{\prime}$ for the triangulation of the $(n+1)$-ball obtained by attaching a new $(n+1)$-cell to $\Sigma $ at the face $\sigma $ corresponding to D. Consider the new compact subspace ${K}^{\prime}$ of $\{\partial {\Sigma}^{\prime},Z\}{\times}_{\{\partial {\Sigma}^{\prime},\underline{Sing}(X)\}}\{{\Sigma}^{\prime},\underline{Sing}(X)\}$: for each element of K, change its image in $\{\Sigma ,\underline{Sing}(X)\}$ by attaching a new $(n+1)$-cell via the map attaching ${D}^{\prime}$ at D, and change its image in $\{\partial \Sigma ,Z\}$ by replacing the image of $\sigma $ with an $(n+1)$-horn via the map attaching ${D}^{\prime}$ at D. Please note that $f({K}^{\prime})$ has fewer fat n-cells in ${Z}^{m}\setminus {Z}^{m-1}$ than $f(K)$. Note too that, if we can find a compact cover ${K}_{1}^{\prime},\dots ,{K}_{j}^{\prime}$ of ${K}^{\prime}$ such that the inclusion map of each ${K}_{i}^{\prime}$ into the pullback lifts to a map ${K}_{i}^{\prime}\to \{{\Sigma}^{\prime},Z\}$, we can use the fact that $Z\to \underline{Sing}(X)$ is a fibration in the compact Hausdorff structure to get a compact cover ${K}_{1},\dots ,{K}_{j}$ of K such that the inclusion map of each ${K}_{i}$ into $\{\partial \Sigma ,Z\}{\times}_{\{\partial \Sigma ,\underline{Sing}(X)\}}\{\Sigma ,\underline{Sing}(X)\}$ lifts to a map ${K}_{i}\to \{\Sigma ,Z\}$.So by applying this procedure finitely many times we reduce to the case where $f(K)$ has no fat n-cells of this type in ${Z}^{m}\setminus {Z}^{m-1}$.
- (ii)
- For each fat n-cell D in case (ii), the approach is similar. Let ${\Sigma}^{\prime}$ be the triangulation given by attaching a new $(n+1)$-cell to $\Sigma $ at the two faces corresponding to D. Consider the new compact subspace ${K}^{\prime}$ of $\{\partial {\Sigma}^{\prime},Z\}{\times}_{\{\partial {\Sigma}^{\prime},\underline{Sing}(X)\}}\{{\Sigma}^{\prime},\underline{Sing}(X)\}$: for each element of K, change its image in $\{\Sigma ,\underline{Sing}(X)\}$ by attaching a new degenerate $(n+1)$-cell coming from the degeneracy maps on D to the faces corresponding to ${\sigma}_{1}$ and ${\sigma}_{2}$, and change its image in $\{\partial \Sigma ,Z\}$ by replacing the images of ${\sigma}_{1}$ and ${\sigma}_{2}$ with the other faces of the new $(n+1)$-cell. These other faces are the degeneracies of $(n-2)$-cells in $f(K)$, so they are in ${Z}^{m-1}$; so, as before, we reduce the number of fat cells in ${Z}^{m}\setminus {Z}^{m-1}$. As before, if we can find a compact cover ${K}_{1}^{\prime},\dots ,{K}_{j}^{\prime}$ of ${K}^{\prime}$ such that the inclusion map of each ${K}_{i}^{\prime}$ into the pullback lifts to a map ${K}_{i}^{\prime}\to \{{\Sigma}^{\prime},Z\}$, we can use the fact that $Z\to \underline{Sing}(X)$ is a fibration in the compact Hausdorff structure to get a compact cover ${K}_{1},\dots ,{K}_{j}$ of K such that the inclusion map of each ${K}_{i}$ into $\{\partial \Sigma ,Z\}{\times}_{\{\partial \Sigma ,\underline{Sing}(X)\}}\{\Sigma ,\underline{Sing}(X)\}$ lifts to a map ${K}_{i}\to \{\Sigma ,Z\}$.So by applying this procedure finitely many times we reduce to the case where $f(K)$ has no fat n-cells in ${Z}^{m}\setminus {Z}^{m-1}$, and we are done.

## 6. Excision

**Proposition**

**4.**

**Proof.**

**Proposition**

**5.**

**Proof.**

**Lemma**

**5.**

**Proof.**

**Theorem**

**9.**

**Proof.**

**Remark**

**3.**

**Theorem**

**10**

**.**Given subspaces $A\subseteq B\subseteq X$ in $\mathcal{U}$ with A closed and B open, the inclusion $(X\setminus A,B\setminus A)\to (X,B)$ induces isomorphisms of the homology group objects ${H}_{n}(X\setminus A,B\setminus A)\to {H}_{n}(X,B)$ for all n. Equivalently, for open subspaces $A,B\subseteq X$ covering X, the inclusion $(B,A\cap B)\to (X,A)$ induces isomorphisms of homology group objects ${H}_{n}(B,A\cap B)\to {H}_{n}(X,A)$ for all n.

**Proof.**

**Theorem**

**11**

**Proof.**

- (i)
- homotopy invariance: homotopies in $\mathcal{U}$ induce homotopies in R-$\mathcal{U}Mod$;
- (ii)
- exactness: associated naturally to a pair $(X,Y)$ is an exact triangle $E(Y,\varnothing )\to E(X,\varnothing )\to E(X,Y)\to $;
- (iii)
- additivity: if $(X,A)$ is a disjoint union of pairs $(\u2a06{X}_{i},\u2a06{A}_{i})$, then the canonical map $\u2a01E({X}_{i},{A}_{i})\to E(X,A)$ is a weak equivalence;
- (iv)
- dimension: $E(*,\varnothing )$ is exact in non-zero dimensions;
- (v)
- excision: for $A\subseteq B\subseteq X$ with A closed and B open, the canonical map $E(X\setminus A,B\setminus A)\to E(X,A)$ is a weak equivalence.

**Theorem**

**12.**

## 7. Eilenberg–Mac Lane Spaces

**Lemma**

**6.**

**Proof.**

**Proposition**

**6.**

**Proof.**

**Theorem**

**13.**

**Proof.**

**Theorem**

**14.**

**Example**

**2.**

**Proposition**

**7.**

**Proof.**

## 8. Conclusions

## Funding

## Conflicts of Interest

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Corob Cook, G.
Eilenberg–Mac Lane Spaces for Topological Groups. *Axioms* **2019**, *8*, 90.
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Corob Cook G.
Eilenberg–Mac Lane Spaces for Topological Groups. *Axioms*. 2019; 8(3):90.
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Corob Cook, Ged.
2019. "Eilenberg–Mac Lane Spaces for Topological Groups" *Axioms* 8, no. 3: 90.
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