Math Problems Vault

Problem 6.8.1

Use Theorem 6.6.2 (a homotopy equivalence of pairs induces an isomorphism in homology) and the dimension property to show that

H_k(D^n) = \begin{cases} \mathbb {Z} & k = 0, \\ 0 & k > 0. \end{cases}

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Problem 6.8.2

Show that

H_k(S^0) = \begin{cases} \mathbb {Z} \oplus \mathbb {Z} & k = 0, \\ 0 & k > 0. \end{cases}

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Problem 6.8.3

Use the long exact sequence of the pair $(D^1, S^0)$ to show that

H_k(D^1, S^0) = \begin{cases} \mathbb {Z} & k = 1, \\ 0 & k \neq 1. \end{cases}

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Problem 6.8.4

Give an explicit singular simplex $\sigma : \Delta_1 \to D^1$ which represents the generator of $H_1(D^1, S^0)$ (as computed in Exercise 6.8.3).

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Remark.

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It should map via $\partial$ to a generator of the kernel of $i_*^0 : H_0(S^0) \to H_0(D^1)$ .

Problem 6.8.5

Show that there is a homeomorphism $h : D^1 \to S^1_+ = \left\{ x \in S^1 : x_2 \geq 0\right\}$ and use this to compute $H_k(S^1_+, S^0)$ for all $k \geq 0$ .

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Problem 6.8.6

(a)(b)

Suppose $B = S^1_+ \subset S^1$ is the upper semicircle $x_2 \geq 0$ and $A = S^1_-$ is the lower semicircle.

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(a)

Use Exercise 6.7.7 to show that there is an isomorphism $H_k(S^1_+, S^0) \to H_k(S^1, S^1_-)$ . Indicate how to find the sets $U, U'$ used in that exercise.

(b)

Use Exercise 6.7.8 to show that there is an isomorphism $H_k(S^1_+, S^0) \to H_k(S^1, S^1_-)$ . Indicate how to find the required bicollar neighborhood.

Problem 6.8.7

Use Exercises 6.8.5 and 6.8.6 to compute $H_k(S^1, S^1_-)$ .

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Problem 6.8.8

Give an explicit singular simplex $\sigma : \Delta_1 \to S^1$ which represents a generator of $H_1(S^1, S^1_-)$ (as computed in Exercise 6.8.7).

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Remark.

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The essential ingredient of a generator is that it corresponds to a generator of $H_1(S^1_+, S^0)$ , and this generator must map to a generator of $\ker (i_*^0) \subset H_0(S^0)$ . From earlier work (Exercise 6.8.4), $\ker (i_*^0)$ is generated by $([1] - [-1])$ . There is a simple trigonometric formula that works for $\sigma$ .

Problem 6.8.9

Use Exercises 6.8.7 and 6.8.8 to compute $H_k(S^1)$ by using the long exact sequence of the pair $(S^1, S^1_-)$ .

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Problem 6.8.10

Give an explicit singular chain which is a sum of two singular simplices, one of which is the answer to Exercise 6.8.8 and the other is a singular simplex in $S^1_-$ , which represents a generator of $H_1(S^1)$ . Explain how this relates to the isomorphism between $H_1(S^1)$ and $\pi_1(S^1, 1)$ .

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Problem 6.8.11

Suppose $p \geq 1$ . Show that $H_{k+1}(D^{p+1}, S^p) \simeq H_k(S^p)$ , $k > 0$ . Show that $H_1(D^{p+1}, S^p) = 0$ .

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Remark.

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Exercises 6.8.1--6.8.10 establish a chain of isomorphisms $H_{k+1}(S^1) \simeq H_{k+1}(S^1, S^1_-) \simeq H_{k+1}(S^1_+, S^0) \simeq H_{k+1}(D^1, S^0)$ . When $k > 0$ , there is also an isomorphism $H_{k+1}(D^1, S^0) \simeq H_k(S^0)$ . When $k = 0$ , the argument is slightly different since $S^0$ is not path connected, and instead one uses the isomorphism $H_1(D^1, S^0) \simeq \ker (i_*^0)$ . This exercise and the ones that follow generalize these arguments to compute the homology of $S^p$ and $(D^{p+1}, S^p)$ for $p \geq 1$ via an inductive argument. Since $S^p, D^{p+1}$ are path connected, $H_0(S^p) \simeq H_0(D^{p+1}) \simeq \mathbb {Z}$ , and the portion of the exact sequence $H_0(S^p) \to H_0(D^{p+1}) \to H_0(D^{p+1}, S^p) \to 0$ shows that $H_0(D^{p+1}, S^p) = 0$ , since the first map is an isomorphism.

Problem 6.8.12

Suppose $p \geq 1$ . Show that $H_k(D^{p+1}, S^p) \simeq H_k(S^{p+1}_+, S^p)$ , $k \geq 0$ . Here $S^{p+1}_+$ denotes the part of the sphere where the last coordinate is $\geq 0$ .

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Problem 6.8.13

Suppose $p \geq 1$ . Show that $H_k(S^{p+1}_+, S^p) \simeq H_k(S^{p+1}, S^{p+1}_-)$ , $k \geq 0$ .

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Problem 6.8.14

Suppose $p \geq 1$ . Show that $H_k(S^{p+1}) \simeq H_k(S^{p+1}, S^{p+1}_-)$ , $k \geq 1$ .

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Problem 6.8.15

Suppose $p \geq 1$ . Combine Exercises 6.8.11--6.8.14 to show that $H_{k+1}(S^{p+1}) \simeq H_k(S^p)$ for $k \geq 1$ , and $H_1(S^{p+1}) = 0$ .

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Problem 6.8.16

(a)(b)

Use an inductive argument (based on Exercises 6.8.11--6.8.15) to show that

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(a)

H_k(S^n) = \begin{cases} \mathbb {Z} \oplus \mathbb {Z} & n = k = 0, \\ \mathbb {Z} & n = k > 0 \text{ or } n > 0, k = 0, \\ 0 & \text{otherwise}; \end{cases}

(b)

H_k(D^n, S^{n-1}) = \begin{cases} \mathbb {Z} & k = n \geq 0, \\ 0 & \text{otherwise}. \end{cases}

Problem 6.8.17

If $A \subset X$ with inclusion map $i$ , a map $r : X \to A$ is called a retraction if $ri = 1_A$ . Show that, if $r : X \to A$ is a retraction, then $r_* : H_k(X) \to H_k(A)$ is surjective for each $k$ .

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Problem 6.8.18

Show that, if there is a $k$ with $H_k(X) = 0$ and $H_k(A) \neq 0$ , then there is no retraction (as defined in Exercise 6.8.17) of $X$ onto $A$ .

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Problem 6.8.19

Show that $D^n$ does not retract onto $S^{n-1}$ , $n \geq 1$ .

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Problem 6.8.20

Show that if $f : D^n \to D^n$ is continuous, then there is a fixed point $x \in D^n$ with $f(x) = x$ .

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Remark.

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See the proof of Theorem 3.5.4.

Problem 6.8.21

(a)(b)

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(a)

If $M$ denotes the Mobius band with boundary $\partial M$ , show that $H_1(\partial M)$ and $H_1(M)$ are each isomorphic to $\mathbb {Z}$ , but the induced map $i_*$ is given by multiplication by $2$ .

(b)

Use (a) to show that a Mobius band does not retract onto its boundary even though they have isomorphic homology.