This will be due March 21.

A. Let \(A\) and \(B\) be abelian groups and \(f\) a homomorphism. Show that \(0\to A\xrightarrow{f} B\) is exact if and only if \(f\) is injective.

B. Suppose \(\cdots \to A_{n+1} \xrightarrow{f_{n+1}} A_n \xrightarrow{f_n} A_{n-1} \to \cdots\) is a long exact sequence. Prove that for each \(n\) there exists a short exact sequence \(0\to\mathrm{coker\ } f_{n+2} \xrightarrow{\phi} A_n \xrightarrow{\theta} \mathrm{ker\ }f_{n-1} \to 0\), where \(\phi\) is induced from \(f_{n+1}\) and \(\theta\) is induced from \(f_n\).

C. Suppose \(A \subset X\), with \(i:A\to X\) the inclusion map, and suppose there is a retraction \(r:X\to A\). Prove that the induced map \(i_\ast: H_n(A)\to H_n(X)\) is a monomorphism and that the induced map \(r_\ast: H_n(X)\to H_n(A)\) is an epimorpishm, for all \(n\).

D. Let \(X\) be a topological space and suppose \(f:X\to X\) is a constant map. Show that the induced map on reduced homology \(f_\ast:\widetilde{H}_n(X)\to \widetilde{H}_n(X) \) is the zero map for all \(n\).

E. Suppose \(A\) is an abelian group and that \(0 \to \mathbb{Z} \xrightarrow{\times 5} \mathbb{Z} \to A \to \mathbb{Z} \to \mathbb{Z} \to 0\) is exact. Classify \(A\).

F. Let \(x_0\) be a point in a space \(X\). Show that \(H_0(X,x_0) \cong \widetilde{H}_0(X)\). Hint: play with the quotient \( Z_0(X,x_0) / B_0(X,x_0) \), and recall the theorem involving \(\widetilde{H}_0(X)\).

Problems G and H relate to the Snake Lemma. Suppose we have the following commutative diagram of abelian groups where the rows are exact:

\( \newcommand{\ra}[1]{\kern-1.5ex\xrightarrow{\ \ #1\ \ }\phantom{}\kern-1.5ex} \newcommand{\ras}[1]{\kern-1.5ex\xrightarrow{\ \ \smash{#1}\ \ }\phantom{}\kern-1.5ex} \newcommand{\da}[1]{\bigg\downarrow\raise.5ex\rlap{\scriptstyle#1}} \begin{array}{c} & & M’ & \ra{f} & M & \ra{g} & M” & \ra{ } & 0 \\ & & \da{d’} & & \da{d} & & \da{d”} & & \\ 0 & \ras{ } & N’ & \ras{f’} & N & \ras{g’} & N” & & \\ \end{array} \)
The Snake Lemma says that the sequence \( \mathrm{ker\ } d’ \xrightarrow{\overline{f}} \mathrm{ker\ } d \xrightarrow{\overline{g}} \mathrm{ker\ } d”\xrightarrow{\partial} \mathrm{coker\ } d’ \xrightarrow{\overline{f’}} \mathrm{coker\ } d \xrightarrow{\overline{g’}} \mathrm{coker\ } d” \) is exact, where \(\partial = (f’)^{-1}\circ d \circ g^{-1}\).

G. Prove the connecting map \(\partial\) is well-defined.

H. Show \(\mathrm{im\ }\partial = \mathrm{ker\ } \overline{f’}\).