Algebraic Topology

Allen Hatcher

Chapter 0

Some Underlying Geometric Notions - all with Video Answers

Educators

Chapter Questions

Problem 1

Construct an explicit deformation retraction of the torus with one point deleted onto a graph consisting of two circles intersecting in a point, namely, longitude and meridian circles of the torus.

Jennifer Stoner

Jennifer Stoner

Numerade Educator

Problem 2

Construct an explicit deformation retraction of $\mathbb{R}^{n}-\{0\}$ onto $S^{n-1}$.

Jennifer Stoner

Jennifer Stoner

Numerade Educator

Problem 3

(a) Show that the composition of homotopy equivalences $X \rightarrow Y$ and $Y \rightarrow Z$ is a homotopy equivalence $X \rightarrow Z .$ Deduce that homotopy equivalence is an equivalence relation.

Chris Trentman

Numerade Educator

Problem 4

A deformation retraction in the weak sense of a space $X$ to a subspace $A$ is a homotopy $f_{t}: X \rightarrow X$ such that $f_{0}=\mathbb{1}, f_{1}(X) \subset A,$ and $f_{t}(A) \subset A$ for all $t .$ Show that if $X$ deformation retracts to $A$ in this weak sense, then the inclusion $A \hookrightarrow X$ is a homotopy equivalence.

Victor Salazar

Numerade Educator

Problem 5

Show that if a space $X$ deformation retracts to a point $x \in X,$ then for each neighborhood $U$ of $x$ in $X$ there exists a neighborhood $V \subset U$ of $x$ such that the inclusion $V \hookrightarrow U$ is nullhomotopic.

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Ed Tam

Numerade Educator

Problem 6

(a) Let $X$ be the subspace of $\mathbb{R}^{2}$ consisting of the horizontal segment $[0,1] \times\{0\}$ together with all the vertical segments $\{r\} \times[0,1-r]$ for $r$ a rational number in $[0,1] .$ Show that $X$ deformation retracts to any point in the segment $[0,1] \times\{0\},$ but not to any other point. ISee the preceding problem.]
(b) Let $Y$ be the subspace of $\mathbb{R}^{2}$ that is the union of an infinite number of copies of $X$ arranged as in the figure below. Show that $Y$ is contractible but does not deformation retract onto any point.
(c) Let $Z$ be the zigzag subspace of $Y$ homeomorphic to $\mathbb{R}$ indicated by the heavier line. Show there is a deformation retraction in the weak sense (see Exercise 4) of $Y$ onto $Z,$ but no true deformation retraction.

Aidan Mcnabb

Numerade Educator

Problem 7

Fill in the details in the following construction from [Edwards $1999]$ of a compact space $Y \subset \mathbb{R}^{3}$ with the same properties as the space $Y$ in Exercise $6,$ that is, $Y$ is contractible but does not deformation retract to any point. To begin, let $X$ be the union of an infinite sequence of cones on the Cantor set arranged end-to-end, as in the figure. Next, form the one-point compactification of $X \times \mathbb{R} .$ This embeds in $\mathbb{R}^{3}$ as a closed disk with curved 'fins' attached along circular arcs, and with the one-point compactification of $X$ as a cross-sectional slice. The desired space $Y$ is then obtained from this subspace of $\mathbb{R}^{3}$ by wrapping one more cone on the Cantor set around the boundary of the disk.

Brian Ketelobeter

Brian Ketelobeter

Numerade Educator

Problem 8

For $n>2,$ construct an $n$ -room analog of the house with two rooms.

Chris Trentman

Numerade Educator

Problem 9

Show that a retract of a contractible space is contractible.

Victor Salazar

Numerade Educator

Problem 10

Show that a space $X$ is contractible iff every map $f: X \rightarrow Y,$ for arbitrary $Y,$ is nullhomotopic. Similarly, show $X$ is contractible iff every map $f: Y \rightarrow X$ is nullhomotopic.

Victor Salazar

Numerade Educator

Problem 11

Show that $f: X \rightarrow Y$ is a homotopy equivalence if there exist maps $g, h: Y \rightarrow X$ such that $f g \simeq \mathbb{1}$ and $h f \simeq \mathbb{1} .$ More generally, show that $f$ is a homotopy equivalence if $f g$ and $h f$ are homotopy equivalences.

Anthony Ramos

Numerade Educator

Problem 12

Show that a homotopy equivalence $f: X \rightarrow Y$ induces a bijection between the set of path-components of $X$ and the set of path-components of $Y,$ and that $f$ restricts to a homotopy equivalence from each path-component of $X$ to the corresponding pathcomponent of $Y$. Prove also the corresponding statement with components instead of path-components. Deduce from this that if the components and path-components of a space coincide, then the same is true for any homotopy equivalent space.

Mohamed Mohamed

Mohamed Mohamed

Numerade Educator

Problem 13

Show that any two deformation retractions $r_{t}^{0}$ and $r_{t}^{1}$ of a space $X$ onto a subspace $A$ can be joined by a continuous family of deformation retractions $r_{t}^{s}$ $0 \leq s \leq 1,$ of $X$ onto $A,$ where continuity means that the map $X \times I \times I \rightarrow X$ sending $(x, s, t)$ to $r_{t}^{s}(x)$ is continuous.

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Problem 14

Given positive integers $v, e,$ and $f$ satisfying $v-e+f=2,$ construct a cell structure on $S^{2}$ having $v$ 0-cells, $e$ 1-cells, and $f$ 2-cells.

Rajesh Singh

Numerade Educator

Problem 15

Enumerate all the subcomplexes of $S^{\infty},$ with the cell structure described in this section, having two cells in each dimension.

ES

Eugene Schneider

University of Minnesota - Twin Cities

Problem 16

Show that $S^{\infty}$ is contractible.

Vishnu P

Numerade Educator

Problem 17

Construct a 2 -dimensional cell complex that contains both an annulus $S^{1} \times I$ and a Möbius band as deformation retracts.

Jennifer Stoner

Jennifer Stoner

Numerade Educator

Problem 18

Show that $S^{1} * S^{1}=S^{3},$ and more generally $S^{m} * S^{n}=S^{m+n+1}$.

Wendi Zhao

Numerade Educator

Problem 19

Show that the space obtained from $S^{2}$ by attaching $n$ 2-cells along any collection of $n$ circles in $S^{2}$ is homotopy equivalent to the wedge sum of $n+1$ 2-spheres.

Uma Kumari

Numerade Educator

Problem 20

Show that the subspace $X \subset \mathbb{R}^{3}$ formed by a Klein bottle intersecting itself in a circle, as shown in the figure, is homotopy equivalent to $S^{1} \vee S^{1} \vee S^{2}$.

E R

Numerade Educator

Problem 21

If $X$ is a connected space that is a union of a finite number of 2 -spheres, any two of which intersect in at most one point, show that $X$ is homotopy equivalent to a wedge sum of $S^{1}$ 's and $S^{2}$ 's.

Regina Hays

Numerade Educator

Problem 22

Let $X$ be a finite graph lying in a half-plane $P \subset \mathbb{R}^{3}$ and intersecting the edge of $P$ in a subset of the vertices of $X .$ Describe the homotopy type of the 'surface of revolution' obtained by rotating $X$ about the edge line of $P$.

Fuzail Shakir

Numerade Educator

Problem 23

Show that a CW complex is contractible if it is the union of two contractible subcomplexes whose intersection is also contractible.

Victor Salazar

Numerade Educator

Problem 24

Let $X$ and $Y$ be $C W$ complexes with 0 -cells $x_{0}$ and $y_{0}$. Show that the quotient spaces $X * Y /\left(X *\left\{y_{0}\right\} \cup\left\{x_{0}\right\} * Y\right)$ and $S(X \wedge Y) / S\left(\left\{x_{0}\right\} \wedge\left\{y_{0}\right\}\right)$ are homeomorphic,
and deduce that $X * Y \simeq S(X \wedge Y)$.

Anthony Ramos

Numerade Educator

Problem 25

If $X$ is a CW complex with components $X_{\alpha},$ show that the suspension $S X$ is homotopy equivalent to $Y \vee_{\alpha} S X_{\alpha}$ for some graph $Y .$ In the case that $X$ is a finite graph, show that $S X$ is homotopy equivalent to a wedge sum of circles and 2 -spheres.

R M

Numerade Educator

Problem 26

Use Corollary 0.20 to show that if $(X, A)$ has the homotopy extension property, then $X \times I$ deformation retracts to $X \times\{0\} \cup A \times I .$ Deduce from this that Proposition 0.18 holds more generally when $(X, A)$ satisfies the homotopy extension property.

Harshita Goel

Numerade Educator

Problem 27

Given a pair $(X, A)$ and a map $f: A \rightarrow B,$ define $X / f$ to be the quotient space of $X$ obtained by identifying points in $A$ having the same image in $B$. Show that the quotient map $X \rightarrow X / f$ is a homotopy equivalence if $f$ is a surjective homotopy equivalence and $(X, A)$ has the homotopy extension property. [Hint: Consider $X \cup M_{f}$ and use the preceding problem.] When $B$ is a point this gives another proof of Proposition 0.17. Another interesting special case is when $f$ is the projection $A \times I \rightarrow A$.

Anthony Ramos

Numerade Educator

Problem 28

Show that if $\left(X_{1}, A\right)$ satisfies the homotopy extension property, then so does every pair $\left(X_{0} \sqcup_{f} X_{1}, X_{0}\right)$ obtained by attaching $X_{1}$ to a space $X_{0}$ via a map $f: A \rightarrow X_{0}$.

Harshita Goel

Numerade Educator

Problem 29

In case the CW complex $X$ is obtained from a subcomplex $A$ by attaching a single cell $e^{n},$ describe exactly what the extension of a homotopy $f_{t}: A \rightarrow Y$ to $X$ given by the proof of Proposition 0.16 looks like. That is, for a point $x \in e^{n},$ describe the path $f_{t}(x)$ for the extended $f_{t}$.

Hast Aggarwal

Numerade Educator