Singular versus Simplicial Homology of a Point; $\pi_0(X,x_0)$ versus $H_0(X,x_0)$ and $\pi_1(X,x_0)$ versus $H_1(X,x_0)$

Let $X$ be a topological space with chosen basepoint $x_0 \in X$. Then $\pi_0(X,x_0)$ is the set (groupoid) of equivalence classes of homotopy equivalence classes of maps from $(S^0, 1)$ to $X$ such that $1$ maps to $x_0$. This is just the set of path components of $X$, so that $X=\{pt\}$ gives that $\pi_0(\{pt\},\{pt\}) = 0$ [there is only one equivalence class of paths here]. Simplicial homology of a $\Delta$-complex $X$ [Hatcher] are given by $\Delta_n(X)$ the free abelian group with basis the open $n$-simplices $e_\alpha^n$ of $X$, and differential given by the boundary map \[ \partial_n(\sigma_\alpha) = \sum_i(-1)^i \sigma_\alpha|[v_0, \cdots, \hat{v_i}, \cdots, v_n] \] Viewing $X=\{pt\}$ as a $\Delta$ complex gives one $0$-cell, so that the simplicial chain complex is given by \[ 0 \rightarrow 0 \rightarrow \cdots \rightarrow 0 \rightarrow \mathbb{Z} \rightarrow 0 \cdots \] with $\mathbb{Z}$ in homological dimension $0$. Now, simplicial homology here gives \[H_i(\{pt\}) = \begin{cases} \mathbb{Z} & \text{ } i=0, \\ 0 & \text{ else }. \end{cases} \] In this sense, $\pi_0(X,x_0)$ does not always ``contain" $H_0(X, x_0)$. Since $H_1(X,x_0) = [\pi_1(X,x_0),\pi_1(X,x_0)]$ is the commutator of $\pi_1(X,x_0)$ for $X$ path-connected, $\pi_1(X,x_0)$ always contains a subgroup isomorphic to $H_1(X,x_0)$ whenever $X$ is path-connected. Singular homology, on the other hand, constructs $n$-chains $C_n(X)$ given by all continuous maps of $n-$simplices to a topological space $X$: $\sigma: \Delta^n\rightarrow X$ with boundary map $\partial_n: C_n(X) \rightarrow C_{n-1}(X)$ given by \[ \partial_n(\sigma) = \sum_i (-1)^i \sigma|[v_0,\cdots,\hat{v_i},\cdots,v_n] \] Here we see that the singular complex of a point $C_\bullet(\{pt\})$ with coefficients in $\mathbb{Z}$ has $\mathbb{Z}$ in $\bf{every}$ non-negative homological index: $C_i(\{pt\}, \mathbb{Z}) = \mathbb{Z}$ for $i\geq 0$, and the boundary map is $0$ in odd homological index [the boundary of an odd-dimensional simplex has an even number of facets, which cancel here], with image $\partial_n(C_n(\{pt\}) = \mathbb{Z}$ for $n$ even. That is: \[ C_\bullet(\{pt\}) = \cdots \rightarrow \mathbb{Z} \stackrel{\partial_{m+1}}{\rightarrow} \mathbb{Z} \stackrel{\partial_m}{\rightarrow} \mathbb{Z} \stackrel{\partial_{m-1}}{\rightarrow} \mathbb{Z} \rightarrow \cdots \rightarrow \mathbb{Z} \stackrel{\partial_1}{\rightarrow} \mathbb{Z} \stackrel{\partial_0}{\rightarrow} 0 \] If $m$ is even here, for example, then $\partial_m(C_m(\{pt\}) = \mathbb{Z}$, and $\partial_{m+1}(C_m(\{pt\}) = \partial_{m-1}(C_{m-1}(\{pt\}) = 0$ so that singular homology gives \[ H_i(\{pt\},\mathbb{Z}) = \begin{cases} \mathbb{Z} & \text{ }i=0, \\ 0 & \text{ else } \end{cases} \] as above for simplicial homology. \[ \] \[ \] \[ \] References \[ \] Hatcher, A. (2002). Algebraic topology. Cambridge: Cambridge University Press. ISBN: 0-521-79160-X; 0-521-79540-0