# HW12

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Homework Assignment 12

Required reading. Read, reread and rereread your notes to this point, and make sure that you really, really really, really really really understand everything in them. Do the same every week! Also, lightly read Hatcher's treatment of cohomology, just enough to get a general impression of that subject.

Solve the following problems. (But submit only the underlined ones). In Hatcher's book problems 5, 6, 8, 9 on page 205.

Due date. Place your assignment in my mailbox by Tuesday April 18th at noon. Put it in the brown envelope marked "Topology TE12" which will be placed there on the last day of classes.

 Above the line, edit only if you are sure Feel free to edit below the line

## Problems for Submission

### Hatcher $\S$3.1, p. 205

5.
Regarding a cochain $\varphi\in C^1(X;G)$ as a function from paths in X to G, show that if $\varphi$ is a cocycle, then
• $\varphi(f\cdot g)=\varphi(f)+\varphi(g)$,
• $\varphi$ takes the value 0 on constant paths,
• $\varphi(f)=\varphi(g)$ if $f\simeq g$,
• $\varphi$ is a coboundary iff $\varphi(f)$ depends only on the endpoints of f, for all f.
(In particular, the first and third properties give a map $H^1(X;G)\to \operatorname{Hom}(\pi_1(X),G)$, which the universal coefficient theorem says is an isomorphism if X is path-connected.)
6.
Directly from the definitions, compute the simplicial cohomology groups of $S^1\times S^1$ with $\mathbb{Z}$ and $\mathbb{Z}_2$ coefficients, using the Δ-complex structure given in $\S$2.1. Do the same for $\mathbb{RP}^2$ and the Klein bottle.