Topology General Exam

August 24, 2012

Name:

Instructions: This is a four hour exam and ’closed book’. There are eight problems.

Problem 1

(a) Suppose that $t \in \mathbb{R}$ is a regular value of a smooth map $f: \mathbb{R}^{n} \rightarrow \mathbb{R}$, and let $M=f^{-1}(t)$. Explain why $M$ has a nowhere vanishing normal vector field.
(b) If $f(x, y, z)=x^{2}+y^{2}+z^{2}$, check that the hypothesis of part (a) holds when $t=1$, and then draw a picture illustrating the conclusion.

Problem 2

Rigorously prove that the Möbius band is non-orientable.

Problem 3

(a) Let $M$ and $N$ be smooth connected closed (= compact without boundary) manifolds of the same dimension. Show that a submersion $f: M \rightarrow N$ will then be a finite sheeted covering map. (a submersion = a map whose differential is surjective at each point.)
(b) Explain why if $M$ is a connected closed surface, and $f: M \rightarrow S^{2}$ is a submersion, then $f$ must, in fact, be a diffeomorphism.
(c) Explain why if $M$ is a connected closed surface, and $f: M \rightarrow S^{1} \times S^{1}$ is a submersion, then $M$ must be $S^{1} \times S^{1}$.

Problem 4

Let $S^{2} \stackrel{p_{1}}{\longleftrightarrow} S^{2} \vee S^{2} \xrightarrow{p_{2}} S^{2}$ be the two ’projection maps’: the other sphere is collapsed to the basepoint. Then say that a map $f: S^{2} \rightarrow S^{2} \vee S^{2}$ has type ( $m, n$ ) if the degree of $p_{1} \circ f$ is $m$ and the degree of $p_{2} \circ f$ is $n$. Let $X_{f}=\left(S^{2} \vee S^{2}\right) \cup_{f} D^{3}$.
(a) Compute the homology groups of $X_{f}$ if $f$ has type $(4,6)$, describing the homology groups as direct sums of cyclic groups, as usual.
(b) More generally, describe the homology groups of $X_{f}$ if $f$ has type $(m, n)$.

Problem 5

Suppose that $X$ is the union of open sets $X_{1}$ and $X_{2}$, and $Y$ is the union of open sets $Y_{1}$ and $Y_{2}$. Let $f: X \rightarrow Y$ be a map that restricts to maps $f_{1}: X_{1} \rightarrow Y_{1}$ and $f_{2}: X_{2} \rightarrow Y_{2}$, and thus also $f_{12}: X_{1} \cap X_{2} \rightarrow Y_{1} \cap Y_{2}$. Prove that, if $f_{1}, f_{2}$ and $f_{12}$ all induce isomorphisms in homology, then $f_{*}: H_{*}(X) \rightarrow H_{*}(Y)$ will also be an isomorphism.

Problem 6

Suppose $p: \tilde{Y} \rightarrow Y$ is a double cover. If $X$ is a space such that $H_{1}(X)$ is a finite group of odd order, show that any map $f: X \rightarrow Y$ lifts through $p$ : there exists $\tilde{f}: X \rightarrow \tilde{Y}$ such that $f=p \circ \tilde{f}$. (You can assume that $X$ is locally 'friendly'.)

Problem 7

Let $M_{2}(\mathbb{R})$ be the vector space of all $2 \times 2$ real matrices, and let $f: M_{2}(\mathbb{R}) \rightarrow \mathbb{R}$ be given by $f(A)=\operatorname{det}(A)$. The differential of $f$ at $A \in M_{2}(\mathbb{R})$ is a linear map $d_{A} f: M_{2}(\mathbb{R}) \rightarrow \mathbb{R}$.
(a) Compute $d_{A} f(A)$.
(b) Show that $S L_{2}(\mathbb{R})$, the group of $2 \times 2$ real matrices with determinant 1 , is a smooth submanifold of $M_{2}(\mathbb{R})$.
(c) Show that $T_{I} S L_{2}(\mathbb{R})$, the tangent space of $S L_{2}(\mathbb{R})$ at the identity matrix $I$, is the subspace of $M_{2}(\mathbb{R})$ consisting of matrices with trace equal to 0.

Problem 8

Recall that the Brower Fixed Point Theorem says that every continuous self map of the closed $n$-ball $D^{n}$ has a fixed point.
(a) Prove the theorem using homology.
(b) Prove the theorem using the methods of differential topology methods. (Step 1: If a continuous $f$ had no fixed points, a nearby smooth function would also have no fixed points.)