# Construction of Moore Space

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.

• 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. – freakish Feb 12 at 12:28
• Can you explain the construction of $f_{\beta}$? – Mathlover Feb 12 at 12:43

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 $$X$$.