HW10

From 0506Topology

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Homework Assignment 10

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, read Hatcher's pages 97-153 and 166-177.

Solve the following problems. (But submit only the underlined ones). In Hatcher's book pages 131-133, problems 1, 8, 26, 27 and 29 and page 155 problems 3, 4 and 8.

Due date. (postponed!) This assignment is due at the end of our class on Tuesday, March 28, 2006.

Just for fun. Remember the connected sum operation $\#$ from our quick discussion of surfaces? Prove that the connected sum ${\mathbb{R}}{\mathbb{P}}^2\#{\mathbb{R}}{\mathbb{P}}^2\#{\mathbb{R}}{\mathbb{P}}^2$ of three copies of the projetive plane ${\mathbb{R}}{\mathbb{P}}^2$ is homeomorphic to the connected sum ${\mathbb{R}}{\mathbb{P}}^2\# T^2$ of a single projective plane ${\mathbb{R}}{\mathbb{P}}^2$ and single torus $T 2$ .

Above the line, edit only if you are sure

Feel free to edit below the line

Table of contents

1 Solution of "Just for fun" problem:

2 Problems for Submission

2.1 Hatcher , pp. 132-133
2.2 Hatcher , p. 155

[edit]

Solution of "Just for fun" problem:

Here is a solution of the problem, posted by User:Pablo:

[edit]

Problems for Submission

[edit]

Hatcher $\S2.1$ , pp. 132-133

8.

An example of a lens space constructed out of 5 tetrahedra.

Construct a 3-dimensional

Δ

-complex

X

from

n

tetrahedra $T_1,\ldots,T_n$ by the following two steps: First arrange the tetrahedra in a cyclic pattern as in the figure, so that each

T i

shares a common vertical face with its two neighbors

T i - 1

and

T i + 1

, subscripts being taken mod

n

. Then identify the bottom face of

T i

with the top face of

T i + 1

for each

i

. Show that the simplicial homology groups of

X

in dimensions 0, 1, 2, 3 are $\mathbb{Z}, \mathbb{Z}_n, 0, \mathbb{Z}$ , respectively. (The space

X

is an example of a lens space.)

27.

Let $f\colon (X,A)\to (Y,B)$ be a map such that both $f\colon X\to Y$ and the restriction $f\colon A\to B$ are homotopy equivalences.

Show that $f_*\colon H_n(X,A)\to H_n(Y,B)$ is an isomorphism for all $n$ .
For the case of the inclusion $f\colon (D^n,S^{n-1})\to (D^n,D^n-\{0\})$ , show that $f$ is not a homotopy equivalence of pairs—there is no $g\colon (D^n,D^n-\{0\})\to (D^n,S^{n-1})$ such that $f g$ and $g f$ are homotopic to the identity through maps of pairs. (Hint: Observe that a homotopy equivalence of pairs $(X,A)\to (Y,B)$ is also a homotopy equivalence for the pairs obtained by replacing $A$ and $B$ by their closures.)

29.: Show that $S^1\times S^1$ and $S^1\vee S^1 \vee S^2$ have isomorphic homology groups in all dimensions, but their universal covering spaces do not.

[edit]

Hatcher $\S2.2$ , p. 155

3.

Let $f\colon S^n\to S^n$ be a map of degree zero. Show that there exist points $x, y\in S^n$ with $f (x) = x$ and $f (y) = - y$ .
Use this to show that if $F$ is a continuous vector field defined on the unit ball $D n$ in $\mathbb{R}^n$ such that $F(x)\ne 0$ for all $x$ , then there exists a point on $\partial D^n$ where $F$ points radially outward and another point on $\partial D^n$ where $F$ points radially inward.

8.: A polynomial $f (z)$ with complex coefficients, viewed as a map $\mathbb{C}\to\mathbb{C}$ , can always be extended to a continuous map of one-point compactifications $\hat f\colon S^2\to S^2$ . Show that the degree of $\hat f$ equals the degree of $f$ as a polynomial. Show also that the local degree of $\hat f$ at a root of $f$ is the multiplicity of the root.

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HW10

From 0506Topology

Solution of "Just for fun" problem:

Problems for Submission

Hatcher $\S2.1$ , pp. 132-133

Hatcher $\S2.2$ , p. 155

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