Exercises

1. Prove that a homeomorphism $E^n$ onto itself maps each boundary point of $E^n$ to a boundary point.
2. Determine the homology groups of the Klein's bottle.
3. Determine the homology groups of the double torus.
4. Establish the isomorphism $H_0(U\cap V) \longrightarrow H_0(U)\oplus H_0(V)$ in the proof of theorem (35.4)
5. Let $C_k$ be the disjoint union of $k$ copies of $S^1$ in $\mathbb{R}^3$. Determine the homology groups of the complement $\mathbb{R}^3 - C_k$.
6. Determine the homology groups of $\mathbb{R}P^3$. Try computing the homology groups of $\mathbb{R}P^4$.
7. Determine the homology groups of $S^n\vee S^m$. Use exercise 4 of lecture 25. to calculate the homology groups of $S^2\times S^4$.

in

Lecture - XXXVI Maps of spheres

in We are now in a position to prove the general Brouwer's fixed point theorem as well as a few other surprising results concerning maps of spheres. As demonstrated in lecture 10, these higher dimensional analogues were inaccessible via the theory of the fundamental group. We shall introduce the notion of the degree of a map of spheres generalizing the notion introduced in lectures (12-13).