Exercises

1. Verify the displayed results for $\partial\sigma_1$ and $\partial\sigma_2$ in lemma (32.2).
2. By writing out the boundary formula in detail verify equations (32.5) and (32.6).
3. Prove lemma (32.5).
4. Verify the naturality of $\Pi_X$ by proving that the diagram (32.1) commutes.
5. Determine the first homology group of the Klein's bottle.
6. Determine the first homology groups of all the spaces described in the exercises to lecture 26.

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Lecture - XXXIII Homotopy invariance of homology

in Homotopy of maps is one of the most important notions in topology and it is of interest to know what is its effect on the induced maps in homology. The result is simple and direct namely, if $f : X \longrightarrow Y$ and $g:X\longrightarrow Y$ are a pair of homotopic maps then they induce the same maps in homology in every dimension. The further advantage here is that no base points are involved unlike the situation encountered in lecture 11 with the fundamental group. However the proof is not direct as one must algebraize the notion of homotopy in the context of chain maps. This leads to the notion of chain homotopy that we first define. We establish the purely algebraic result that a pair of chain homotopic maps induce equal maps in homology. We then proceed to relate the topological notion of homotopy of a pair of continuous maps $f, g$ as above with the chain homotopy between the induced chain maps $f_{\sharp}:S_n(X)\longrightarrow S_n(Y)$ and $g_{\sharp}:S_n(X)\longrightarrow S_n(Y)$. Some of these ideas have been implicitly used in the last lecture in the construction of the singular two chain $\sigma$ in lemma (32.2). We shall follow the treatment in the book by T. Dieck⁷ defining first the notion of the cross product which seems more transparent. The student who is familiar with differential forms may notice some similarities with wedge products and the exterior derivative. As in the theory of differential forms where the construction of the exterior derivative $d$ is forced upon us through some of its properties, the cross product is determined by its properties described in theorem (33.1), as soon as one chooses for each pair $(p, q)$ a model chain namely, the $p+q$ chain $z$ in (33.4).