Exercises

1. Sketch $\Delta_n$ for $n = 1, 2, 3$. Show that $\Delta_n$ is a compact and connected subspace of $\mathbb{R}^{n+1}$.
2. Discuss the continuity of the maps (29.1). Prove lemma (29.1). what about the cases $i \leq j$?
3. Verify equation (29.8).
4. Determine the values of $n$ ( $n = 1, 2, \dots$) for which a constant function $\Delta_n\longrightarrow X$ an $n-$cycle.
5. Show that the family of all chain complexes forms a category in which the set of morphisms Mor$(G, K)$ between any two chain complexes $G$ and $K$ is the set of all chain maps from $G$ to $K$.
6. Naturality of (29.17)-(29.18). Assume given a commutative diagram of chain complexes with exact rows:
   $\begin{CD} 0@> >> L @> f >> G @> g >> K @> >> 0\\ @. @VV{\phi}V @VV{\psi} V... ...me} @> f^{\prime} >> G^{\prime} @> g^{\prime} >> K^{\prime} @> >> 0\\ \end{CD}$
   Denoting by $\delta_n$ and $\delta_n^{\prime}$ the connecting homomorphisms, sketch relevant diagrams and prove that
   $$\delta_n^{\prime}\circ H_n(\eta) = H_n(\psi)\circ \delta_n$$

in

Lecture - XXXI The homology groups and their functoriality

in Having laid the algebraic foundations in the previous lecture we shall formally define the homology functors $H_n$, $n = 0, 1, 2,\dots$ from the category Top to the category AbGr. We shall discuss $H_0(X)$ completely and show that $H_0(X)$ is free abelian of rank equal to the number of path components of $X$. The groups $H_n(X)$ ($n\geq 1$) vanish when $X$ is a convex subset of $\mathbb{R}^n$. We shall prove this result using a technique that would be be considerably generalized in lecture 33. However the special case proved here for convex subsets would be needed in lecture 33. In the next lecture we shall see examples of topological spaces $X$ for which $H_1(X)$ is non-trivial. However the reader would have to wait till lecture 34 to see more interesting examples.