Math 440, Fall 2014, Assignment 14

I must study politics and war that my sons may have liberty to study mathematics and philosophy.

- John Adams, letter to Abigail Adams, May 12, 1780

Note: this assignment, on homology and cohomology, does not appear on the final exam.

Carefully define the following terms, then give one example and one non-example of each:

1. Singular $n$-cube (in a topological space $X$).
2. Degenerate $n$-cube.
3. Singular $n$-chain.
4. Face (of a singular cube).
5. Boundary operator (on $n$-chains).
6. Singular $n$-cycle.
7. Singular $n$-boundary.
8. The homology group $H_n(X)$.
9. Homomorphism on homology induced by a continuous map.
10. Dual group (of an abelian group).
11. Homomorphism on duals induced by an abelian group homomorphism.
12. Singular $n$-cochain.
13. Coboundary operator.
14. Singular $n$-cocycle.
15. Singular $n$-coboundary.
16. The cohomology group $H^n(X)$.
17. The cohomology ring $H^*(X)$.

Carefully state the following theorems (you do not need to prove them):

1. Homology of $\mathbb{R}^n$, the torus $T^2$, the sphere $S^2$, and the projective plane $\mathbb{RP}^2$.

Solve the following problems:

1. Show that a continuous map $f:X\rightarrow Y$ induces a group homomorphism $\check{f}:H^n(Y)\rightarrow H^n(X)$ (note the reversed arrow!). (In fact, one can paste these morphisms together to get a ring homomorphism $\check{f}:H^*(Y)\rightarrow H^*(X)$. You do not need to prove this.)

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