Homology from definitions
due by Monday, Feb 10, 2020
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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$X$.
- (a). Show that
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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.
- (a). Show that if
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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$n\geq 0$.
- (a). Show that, for