Homology from definitions

due by Monday, Feb 10, 2020
  1. Let $X$ be a topological space and $|X|$ be the set of path-connected components of $X$.

    • (a). Show that $H_0(X)$ is isomorphic to the free abelian group generated by $|X|$.
    • (b). For $n\geq 0$, show that $H_n(X) \cong \oplus_{j\in |X|} H_n(X_j)$, where $\{X_j\}_{j \in |X|}$ are the connected components of $X$.
  2. Recall that the augmented chain complex for a topological space $X$ has $C_{-1}(X) = \mathbb{Z}$ and $\partial_{-1} = 0$.

    • (a). Show that if $X\neq\phi$, there is a short exact sequence $0 \to \widetilde{H_0}(X) \to H_0(X) \to \mathbb{Z}$ .
    • (b). Show that (if $X\neq\phi$) the above short exact sequence splits, and hence we have $H_0(X) = \widetilde{H_0}(X)\oplus \mathbb{Z}$.
    • (c). What happens to tbe above two statements if $X = \phi$.
    • (d). What is $H_{-1}(X)$? Consider both the cases when $X$ is empty and non-empty.
  3. Let $X$ be a topological space, $A\neq \phi$ a closed subset and $U\supset A$ be an open set that deformation retracts to $A$. Let $X/A$ denote the quotient space where all points of $A$ are identified.

    • (a). Show that, for $n\geq 0$, $H_n(X, A) \cong H_n(X, U) \cong H_n(X/ A, U /A)$.
    • (b). Conclude that $H_n(X, A) \cong \widetilde{H_n}(X/A)$ for all $n\geq 0$.