Exercises

1. Show that the map defined by (34.1) is the restriction to $\Delta_p$ of an affine map. Note: An affine map is the composition of a linear map and a translation.
2. Suppose $T:\mathbb{R}^{n+1}\longrightarrow \mathbb{R}^{m+1}$ is an affine map such that $T(\Delta_n)\subset \Delta_m$, then $T_{\sharp}$ maps the subgroup $A_p(\Delta_n)$ into $A_p(\Delta_m)$ and is a chain map from the complex $\{A_p(\Delta_n)\}$ to $\{A_p(\Delta_m)\}$. Further prove the following:

   (i)

   If ${\bf b} \in \Delta_n$ and $\sigma \in A_p(\Delta_n)$ then $T_{\sharp}(K_{\bf b}\sigma) = K_{T{\bf b}}(T_{\sharp}\sigma)$.

   (ii)

   If ${\bf b}$ is the barycenter of $\sigma$ then ${\bf b}$ is the barycenter of $T_{\sharp}\sigma$.

   What happens if we consider a degenerate two simplex where the points ${\bf v}_1, {\bf v}_2, {\bf v}_3$ are not affinely independent? Discuss the case of the two simplex $[{\bf v}_1, {\bf v}_2, {\bf v}_2]$.
3. Examine what happens if the term referred to as junk in equation (34.7) is retained.
4. Complete the details of the proof of theorem (34.4).
5. Show that ${\cal B}^k$ is chain homotopic to the identity map. What is the chain homotopy?
6. Suppose that the maps $g$ and $h$ in the exact sequence
   $$\begin{CD} A @> >> B@> g >> C @> h >> D@> >> E \end{CD}$$
   are replaced by the composites
   $$\begin{CD} {\tilde g}:B@> g >> C@> \lambda >> X,\quad \quad {\tilde h}:X@> \lambda^{-1} >> C@> h >> D \end{CD}$$
   the result is again an exact sequence.
7. Fill in the details in the proof of theorem (34.8). See exercise 6 of lecture 29.

in

Lecture - XXXV The Mayer Vietoris sequence and its applications

in The proof of Mayer Vietoris sequence is reminiscent of the Seifert Van Kampen theorem. While the Seifert Van Kampen theorem enables us to relate the fundamental group of a union $U\cup V$ in terms of the fundamental groups of $U, V$ and $U\cap V$, the situation here is slightly more involved. The precise relationship between the homologies of $U, V, U\cap V$ and $U\cup V$ is described in terms of the long exact sequence of theorem (34.7).

As in the Seifert Van Kampen theorem we obtain from a push-out diagram of topological spaces a push out diagram of chain complexes which turns into a short exact sequence of complexes. The corresponding long-exact sequence gives, after an application of the excision theorem of the last lecture, the Mayer Vietoris sequence. It is one of the most efficient tools available for the computation of homology groups. We restate here the theorem for convenience.