Topology General Exam January 2014

Solve the following problems on your own paper. Be sure your solutions are legible and clearly organized. All work should be your own; no outside sources are permitted. You may use without proof standard results from first-semester differential and algebraic topology; where appropriate you should cite theorems by name.

Problem 1

Suppose $X$ is a connected, locally path-connected space whose fundamental group is finite. Prove that every continuous map from $X$ to the torus $T$ is null-homotopic.

Problem 2

Let $T_{1}$ and $T_{2}$ be two copies of the torus $S^{1} \times S^{1}$, and let $f, g: S^{1} \rightarrow S^{1}$ be two maps of degrees 2 and 6 , respectively. Let $x_{0} \in S^{1}$ be a fixed base point. Find the fundamental group of the space

$$X=T_{1} \cup_{F} S^{1} \times[0,1] \cup_{G} T_{2},$$

where $F: S^{1} \times\{0\} \rightarrow T_{1}$ is given by $F(x, 0)=\left(f(x), x_{0}\right)$ and $G: S^{1} \times\{1\} \rightarrow T_{2}$ is given by $G(x, 1)=\left(g(x), x_{0}\right)$.

Problem 3

You are given the following commutative diagram of abelian groups:

Assume that all columns are exact, and that the first two rows are exact. Prove that the third row is exact at $C_{2}$, i.e., that $\operatorname{im}(i)=\operatorname{ker}(j)$. (Note: in this situation the entire third row is exact, but you need not prove this.)

Problem 4

Consider a space $X$ that is the union of two open subsets $U$ and $V$ such that:

• $V$ is contractible

• both $U$ and $U \cap V$ have the homology of a circle.

a) What are all the possibilities for the homology $H_{*}(X ; \mathbb{Z})$?

b) Describe explicit spaces realizing all these possibilities.

Problem 5

Let $M$ be a smooth compact manifold without boundary. Show that there is no submersion (i.e., smooth map whose differential is everywhere surjective) $F: M \longrightarrow \mathbb{R}^{k}$ for any $k>0$.

Problem 6

Show that for any $n \geq 0$ the manifold $M=S^{n} \times \mathbb{R}$ is parallelizable (that is, its tangent bundle is trivial).

Problem 7

Let $M$ be a smooth, closed (compact without boundary) $n$-dimensional submanifold of $\mathbb{R}^{n+1}$, with $0 \notin M$. Prove that there exists a line through 0 in $\mathbb{R}^{n+1}$ which intersects $M$ in finitely many points (or is disjoint from $M$ ).

Problem 8

Let $M$ and $N$ be the subsets of $\mathbb{R}^{3}$ defined by

$$M=\left\{x^{2}+y^{2}+z^{2}=1\right\} \quad N=\left\{x^{2}-y^{2}+z^{2}=c\right\}$$

for a real number $c$. Justify your responses to the following:
a) Determine all values of $c$ for which $M$ and $N$ are submanifolds of $\mathbb{R}^{3}$, and the intersection $M \cap N$ is transverse.
b) Determine all values of $c$ for which $M \cap N$ is a submanifold of $\mathbb{R}^{3}$.

Problem 9

Suppose $f: S^{n} \longrightarrow S^{n}, n \geq 2$, is a smooth map whose differential is injective at each point. Prove that $f$ is a diffeomorphism.