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While reading the construction of Moore space from Hatcher's Algebraic topology on page 143 , I faced the following problem:---

Let $G$ be an abelian group and $0\rightarrow K\rightarrow F\rightarrow G\rightarrow 0$ be a free resolution of $G$ with $\{x_{\alpha}\}$ and $\{y_{\beta}\}$ as basis of $F$ and $K$ respectively such that $y_{\beta}=\sum_{\alpha} d_{\beta \alpha}x_{\alpha}$. Writing $X^n=\lor_{\alpha} S_n^{\alpha}$, we have $H_n(X^n)=F$. Now author constructed maps $\{f_{\beta}:S^n\rightarrow X^n\}_{\beta}$ such that $deg(p_{\alpha}\circ f_{\beta})=d_{\beta \alpha}$ where, $p_{\alpha}:X^n\rightarrow S_{\alpha}^n$ is the projection map. Now let $X$ be obtained from $X^n$ attaching $e_{\beta}^{n+1}$ via $f_{\beta}$. Then $H_n(X)=G$ and $\tilde H_i(X)=0,\forall i\not =n$. Here $n\geq 1$.

I haven't understood the next lines of this problem where $f_{\beta}$ has been constructed. Also I have a dobut in computing the homology group of this space. Any help will be appreciated. Actually I need some rigorous argument for this problem.

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  • $\begingroup$ So the construction is like this in order to apply the cellural boundary formula, right? Together with the fact that homology of wedge sum is a direct product of homologies. $\endgroup$ – freakish Feb 12 at 12:28
  • $\begingroup$ Can you explain the construction of $f_{\beta}$? $\endgroup$ – Mathlover Feb 12 at 12:43
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Given your free resolution of the group $G$, we have $F=\displaystyle\bigoplus_{g\in G} \mathbb{Z}$ and $\{ y_\beta\}$ is a basis of $K\subseteq F$ the kernel of the projection map $F\rightarrow G$. Define, as you said, the CW complex: $$ X^n=\bigvee_{g\in G} S^n. $$ Since $\widetilde{H}_*$ is a reduced homology theory, it is in particular additive. Thus we get: $$ \widetilde{H}_n(X^n)\cong \bigoplus_{g\in G} \widetilde{H}_n(S^n)\cong \bigoplus_{g\in G} \mathbb{Z} = F. $$ For $n\geq 1$, the space $X^n$ is $(n-1)$-connected, so we can apply in particular the Hurewicz theorem which says there is an isomorphism: $$ h:\pi_n(X^n)\stackrel{\cong}\longrightarrow \widetilde{H}_n(X^n)\cong F. $$ Via the above isomorphism, for each basis element $y_\beta$ in $K$, there is a corresponding element $[f_\beta]_*$ in $\pi_n(X^n)$, i.e., a map $S^n\rightarrow X^n$ uniquely defined up to homotopy. This defines a map: $$ f:\bigvee_{\beta} S^n \stackrel{\bigvee f_\beta}\longrightarrow X^n. $$ Your desired CW complex is then obtained as the pushout: \begin{array} $\displaystyle\bigvee_{\beta}S^n &\stackrel{f}{\longrightarrow} & X^n \\ \downarrow & & \downarrow\\ \displaystyle \bigvee_{\beta} D^{n+1} & \longrightarrow & X. \end{array} Notice that the space $X$ is the mapping cone of the map $f$, this should help compute the homology of $X$.

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