Exercises

1. Show that the $p+q-1$ chain on the right hand side of (33.4) is a cycle.
2. Check that $\sigma\times \tau$ as defined by equation (33.5) satisfies (33.1).
3. Show that the product in theorem (33.1) defines a bilinear map $H_p(X)\times H_q(Y)\longrightarrow H_{p+q}(X\times Y)$.
4. Determine explicitly the two/three chain $z$ satisfying (33.4) when

   (i) $p = 1$ and $q = 1$.

   (ii) $p = 1$ and $q = 2$.

   Hint: In the proof of lemma (32.2), we chopped the square into two triangles. When $q = 2$ we need to chop a prism into three pieces and map $\Delta_3$ affinely onto each of them.

5. Use the map $\Pi_{X}$ of the previous lecture to calculate the generators of $H_1(S^1\times S^1)$.
6. Use equation (33.1) to determine the image of the pair of generating one cycles of the previous exercise under the map $H_1(S^1)\times H_1(S^1)\longrightarrow H_2(S^1\times S^1)$.

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Lecture - XXXIV Small simplicies

in Recall that the Goursat lemma in complex analysis is proved by subdividing a triangle into four smaller triangles determined by the midpoints of the sides of the given triangle. The integral over the given triangle is then the sum of the integrals over the four little pieces. Likewise, in the proof of the classical Green's theorem (of which Cauchy's theorem is really a special case) one employs a subdivision into tiny squares. The contributions to the integral from an edge common to a pair of abutting triangles/squares cancel out.

A similar idea underlies the method of small simplicies where we perform a systematic subdivision operation known as barycentric subdivision. The barycentric subdivision enables us to replace a singular chain by a homotopic one in which the constituent singular simplicies are small. A small simplex is one whose image lies in an open set belonging to a prescribed open cover of the space. One achieves this through iterated barycentric subdivisions a process reminiscent of one used in the proof of the Goursat lemma. The fundamental theorem on small simplicies quickly leads us to the two fundamental results on algebraic topology - the excision theorem discussed in lecture 39 and the Mayer Vietoris sequence that we shall derive here and use in the next lecture.