Homology from definitions
due by Monday, Feb 10, 2020

Let
$X$
be a topological space and$X$
be the set of pathconnected 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

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 nonempty.
 (a). Show that if

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