Math Problems Vault

Problem 6.15.1

(a)(b)(c)

Suppose that $M$ is an $n$ -manifold, and $x \in \mathrm{int}\, M$ . Let $h : D^n \to M$ be the restriction of a homeomorphism $h'$ from $\mathbb {R}^{n}$ onto an open set $U'$ in $\mathrm{int}\, M$ so that if $U = h(\mathrm{int}\, D^n)$ , then $x = h(y) \in U$ .

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(a)

Show that

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

(b)

Show that

H_k(M, M \setminus \left\{ x\right\} ) \simeq H_k(D^n, D^n \setminus \left\{ y\right\} ) \simeq \begin{cases} \mathbb {Z} & k = n, \\ 0 & \text{otherwise}. \end{cases}

(c)

Show that the map $H_k(M, M \setminus U) \to H_k(M, M \setminus \left\{ x\right\} )$ is an isomorphism.

Remark.

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For a manifold $M$ and $x \in M$ , we use the shorthand $H_i^{M,A}$ for $H_i(M, M \setminus A)$ . Whenever $A \subset B$ , there is an inclusion $(M, M \setminus B) \subset (M, M \setminus A)$ inducing a homomorphism $r_{B,A} : H_i^{M,B} \to H_i^{M,A}$ ; if $x \in A \subset B$ , then $r_{A,x} r_{B,A} = r_{B,x}$ . Write $r_x$ for the map $H_n(M) \to H_n^{M,x}$ induced by inclusion (or, for a manifold with boundary, $r_x : H_n(M, \partial M) \to H_n^{M,x}$ for $x \in \mathrm{int}\, M$ ).

If $x \in M$ and $U$ is an open set about $x$ homeomorphic to the interior of an embedded $n$ -disk, then $r_{U,x} : H_n(M, M \setminus U) \to H_n(M, M \setminus \left\{ x\right\} )$ is an isomorphism onto a group isomorphic to $\mathbb {Z}$ ; the homeomorphism $h : (D^n, S^{n-1}) \to (M, M \setminus U) \to (M, M \setminus \left\{ x\right\} )$ gives corresponding generators $\mu_U \in H_n(M, M \setminus U)$ , $\mu_x \in H_n(M, M \setminus \left\{ x\right\} )$ as images of a generator $\mu_D$ of $H_n(D^n, S^{n-1})$ .

Definition 6.15.1: an $n$ -manifold $M$ is orientable if there is a choice $\mu_x \in H_n(M, M \setminus \left\{ x\right\} )$ for each $x \in M$ so that $\mu_x$ is a generator of $H_n(M, M \setminus \left\{ x\right\} )$ for each $x$ , and these choices are locally consistent: for each $x \in M$ there is an open set $U$ containing $x$ , homeomorphic to the interior of a disk in an embedded $\mathbb {R}^{n}$ , and a generator $\mu_U \in H_n(M, M \setminus U) \simeq \mathbb {Z}$ so that $r_{U,y}(\mu_U) = \mu_y$ for all $y \in U$ . A choice of $\mu_x$ for all $x$ as above is called a homology orientation (or just an orientation) for $M$ .

Problem 6.15.2

Show that if $D_+^n$ is the upper half disk and $S_+^{n-1}$ is the upper half of the sphere, then $H_k(D_+^n, S_+^{n-1}) \simeq 0$ .

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Problem 6.15.3

Show that a boundary point $x$ and an interior point $y$ of an $n$ -manifold $M$ with boundary are distinguished by their homology groups $H_n(M, M \setminus \left\{ x\right\} ) \simeq 0$ and $H_n(M, M \setminus \left\{ y\right\} ) \simeq \mathbb {Z}$ (using Exercise 6.15.2 and Exercise 6.15.1(b)).

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Problem 6.15.4

Show that if $M$ is an $n$ -manifold, then it is not an $m$ -manifold for $m \neq n$ .

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Problem 6.15.5

Show that if a connected $n$ -manifold $M$ is orientable (Definition 6.15.1, Exercise 6.15.1), then the choice of homology orientation at a fixed $x_0$ determines the choice everywhere.

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Remark.

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Use the fact that it determines it locally and path connectivity.

Problem 6.15.6

(a)(b)(c)(d)

Suppose $M$ is a compact, connected, homology-oriented $n$ -manifold without boundary (Definition 6.15.1). Denote by $\mu \in H_n(M)$ a generator with $\mu \mapsto \mu_z$ under $r_z : H_n(M) \to H_n^{M,z}$ and $\mu_z$ the class in the definition of homology orientation.

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(a)

Show that if $H_t : M \to M$ is an isotopy and $H_0 = \mathrm{id}$ , then $(H_t)_*(\mu ) = \mu$ for all $t \in I$ .

(b)

Suppose that $H_t(x) = y$ . Show that $(H_t)_*(\mu_x) = \mu_y$ .

(c)

Let $h_0, h_1 : D^n \to M$ be embedded disks which are ambient isotopic; that is, $h_1 = H_1 h_0$ , where $H_t$ is an isotopy of $M$ . Let $\mu_D \in H_n(D^n, D^n \setminus \left\{ 0\right\} )$ be the positive generator. If $(h_0)_* \mu_D = \epsilon \mu_{h_0(0)}$ , show that $(h_1)_* \mu_D = \epsilon \mu_{h_1(0)}$ .

(d)

Show that $h_0$ is not ambient isotopic to $h_0 r$ , where $r : D^n \to D^n$ is a reflection.

(If $M$ is a compact, connected, homology-oriented $n$ -manifold with boundary, the same exercise can be repeated virtually word for word using the class $\mu \in H_n(M, \partial M)$ instead.)

Remark.

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For part (b), the commutative diagram

\begin{array}{ccc} H_n(M) & \xrightarrow {(H_t)_*} & H_n(M) \\ \downarrow {r_x} & & \downarrow {r_y} \\ H_n^{M,x} & \xrightarrow {(H_t)_*} & H_n^{M,y} \end{array}

is helpful.

Theorem 6.15.2, the main theorem toward which this section builds (using the notation $H_i^{M,A} = H_i(M, M\setminus A)$ of the remark to Exercise 6.15.1): let $M$ be a connected $n$ -manifold with boundary $\partial M$ , possibly empty. (1) $H_i(M) = 0$ , $i > n$ . (2) If $M$ is compact and nonorientable, or $M$ is not compact, then $H_n(M) = 0$ . (3) If $\partial M = \emptyset$ and $M$ is compact and orientable, then $r_x : H_n(M) \to H_n^{M,x}$ is an isomorphism for all $x \in M$ . (4) If $M$ is compact with $\partial M \neq \emptyset$ , then $H_n(M) = 0$ . (5) If $M$ is compact with $\partial M \neq \emptyset$ and $M$ is orientable, then $r_x : H_n(M, \partial M) \to H_n^{M,x}$ is an isomorphism for each $x \in \mathrm{int}\, M$ .

Problem 6.15.7

(a)(b)

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(a)

Suppose that $n > 0$ . Without using Theorem 6.15.2 (stated in the remark to Exercise 6.15.6), show that $H_n(S^n) \simeq \mathbb {Z}$ with generator $\mu$ implies that $r_x : H_n(S^n) \to H_n^{S^n, x}$ is an isomorphism.

(b)

If $\mu_x = r_x(\mu )$ , show that $\mu_x$ satisfies the definition of a homology orientation (Definition 6.15.1).

Problem 6.15.8

Suppose $H : \mathbb {R}^{n} \times I \to \mathbb {R}^{n} \times I$ is an isotopy. Extend $H$ to a map $\bar{H} : S^n \times I \to S^n \times I$ by regarding $\mathbb {R}^{n}$ as embedded into $S^n$ as $S^n \setminus \left\{ N\right\}$ , where $N = (0,1)$ , and defining $\bar{H}(N,t) = (N,t)$ . Show that $\bar{H}$ is an isotopy.

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Problem 6.15.9

Suppose that $h_0, h_1 : D^n \to \mathbb {R}^{n}$ are embedded disks which are ambient isotopic. By regarding $\mathbb {R}^{n}$ as embedded in $S^n$ as $S^n \setminus \left\{ N\right\}$ and taking the homology orientation of $\mathbb {R}^{n}$ as coming from the homology orientation of $S^n$ (Exercise 6.15.7, Exercise 6.15.8), show that if $(h_0)_*(\mu_D) = \epsilon \mu_{h_0(0)}$ then $(h_1)_*(\mu_D) = \epsilon \mu_{h_1(0)}$ . Conclude that $h_0$ is not ambient isotopic to $h_0 r$ (with $r$ a reflection of $D^n$ , as in Exercise 6.15.6(d)).

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Problem 6.15.10

(a)(b)(c)(d)

Let $G_t : \mathbb {R}^{n} \to \mathbb {R}^{n}$ be an isotopy which is the identity outside of a large disk $B$ , with $G_0 = \mathrm{id}$ .

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(a)

Show that the map $(G_1)_* : H_n^{\mathbb {R}^{n}, \mathrm{int}\, B} \to H_n^{\mathbb {R}^{n}, \mathrm{int}\, B}$ is the identity.

(b)

Use (a) to show that the map $(G_1)_* : H_n^{\mathbb {R}^{n}, x} \to H_n^{\mathbb {R}^{n}, G_1(x)}$ sends the local homology orientation at $x$ determined via a choice of generator $g_B \in H_n^{\mathbb {R}^{n}, \mathrm{int}\, B}$ to the local homology orientation at $G_1(x)$ determined by $g_B$ .

(c)

Show that if $h_0, h_1 : D^n \to \mathbb {R}^{n}$ are ambient isotopic embedded disks with ambient isotopy $G_t$ which is the identity outside of a large disk $B$ (i.e. $G_0 = \mathrm{id}$ , $G_1 h_0 = h_1$ ), and we choose the homology orientation consistent with $h_0$ , then it is also consistent with $h_1$ .

(d)

Show that $h$ and $hr$ are not ambient isotopic (with $r$ a reflection of $D^n$ ).

Problem 6.15.11

Let $G_t : \mathbb {R}^{n} \to \mathbb {R}^{n}$ be an isotopy with $G_0 = \mathrm{id}$ (without assuming $G_t$ is the identity outside a disk, unlike Exercise 6.15.10).

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(a)

Show that if $D_r$ is a disk about the origin in $\mathbb {R}^{n}$ , then there is another disk $D_s$ for $s \geq r$ about the origin so that $G_t(\mathbb {R}^{n} \setminus D_s) \subset \mathbb {R}^{n} \setminus D_r$ for all $t$ .

(b)

Show that the map $(G_1)_* : H_n^{\mathbb {R}^{n}, D_s} \to H_n^{\mathbb {R}^{n}, D_r}$ sends $\mu_s$ to $\mu_r$ (notation for the standard orientation of $\mathbb {R}^{n}$ as in Proposition 6.15.1).

(c)

Use (b) to show that the map $(G_1)_* : H_n^{\mathbb {R}^{n}, x} \to H_n^{\mathbb {R}^{n}, G_1(x)}$ sends the local homology orientation at $x$ determined via a choice of generator $g_s \in H_n^{\mathbb {R}^{n}, D_s}$ to the local homology orientation at $G_1(x)$ determined by $g_r$ .

(d)

Now suppose that $h_0, h_1 : D^n \to \mathbb {R}^{2}$ are ambient isotopic embedded disks (i.e. $G_0 = \mathrm{id}$ , $G_1 h_0 = h_1$ ). Show that if we choose the homology orientation consistent with $h_0$ , then it is also consistent with $h_1$ .

(e)

Show that $h$ and $hr$ are not ambient isotopic.

Remark.

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For part (a): consider the image of $p_1 \bar{H}^{-1}(D_r \times I)$ , where $p_1 : \mathbb {R}^{n} \times I \to \mathbb {R}^{n}$ is projection onto the first coordinate. This is a compact set and so is contained in some $D_s$ .

Proposition 6.15.1 (the standard orientation of $\mathbb {R}^{n}$ ): $\mathbb {R}^{n}$ is oriented by choosing a consistent set of generators $\mu_r \in H_n^{\mathbb {R}^{n}, D_r} \simeq \mathbb {Z}$ so that, if $r_1 < r_2$ , then $r_{D_{r_2}, D_{r_1}}(\mu_{r_2}) = \mu_{r_1}$ . For any $x \in \mathbb {R}^{n}$ , choosing $r$ with $x \in D_r$ and defining $\mu_x = r_{D_r,x}(\mu_r)$ gives an orientation. If $K$ is a compact convex set contained in $D_r$ and $x \in K$ , the inclusion maps $\mathbb {R}^{n} \setminus D_r \to \mathbb {R}^{n} \setminus K \to \mathbb {R}^{n} \setminus \left\{ x\right\}$ are homotopy equivalences inducing isomorphisms for all $i$ ; $r_{D_r,K}, r_{K,x}$ are each isomorphisms with composition the isomorphism $r_{D_r,x}$ , so $H_i^{\mathbb {R}^{n},K} = 0$ if $i \neq n$ and $H_n^{\mathbb {R}^{n},K} \simeq \mathbb {Z}$ . Choosing $\mu_K = r_{D_r,K}(\mu_r)$ , it satisfies $r_{K,x}(\mu_K) = \mu_x$ for each $x \in K$ . Given an orientation $\mu_x$ , the choice of $\mu_x$ for a single $x$ determines the choice of consistent $\mu_r$ for all $r$ , hence a consistent $\mu_K$ whenever $x \in K$ .

Problem 6.15.12

(a)(b)(c)

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(a)

Show that the elements of $U_\alpha$ are locally consistent in the sense that if we have $v_x \in U_\alpha$ with $v_x \in H_n^{M,x}$ and $W \subset U$ is a neighborhood of $x$ homeomorphic to an open disk with $v_W$ the element of $H_n^{M,W} \simeq H_n^{M,x}$ which maps to $v_x$ under this isomorphism, then $r_{W,y}(v_W) \in U_\alpha$ .

(b)

Show that this does provide a basis for a topology.

(c)

Show that $(\hat{M}, \hat{p}, M)$ and $(\tilde{M}, p, M)$ are covering spaces.

Remark.

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Setup: for a connected $n$ -manifold $M$ , form two covering spaces $\hat{M}, \tilde{M}$ of $M$ . The points of $\tilde{M}$ are elements of the group $H_n^{M,x}$ , where $x$ ranges over the points of $M$ ; to be a point of $\hat{M}$ , the element must be a generator of the group. Here "covering space" is used in a more general sense than before, in that $\tilde{M}, \hat{M}$ are not required to be path connected (in fact, the path connectivity of $\hat{M}$ will be shown to be equivalent to nonorientability); the local triviality condition is still required, so these may be thought of as disjoint unions of covering spaces in the earlier sense. There is a natural inclusion $\hat{M} \subset \tilde{M}$ . Define a projection $p : \tilde{M} \to M$ by sending $k \in H_n^{M,x}$ to $x \in M$ , and by restriction $\hat{p} : \hat{M} \to M$ . To topologize $\tilde{M}, \hat{M}$ so that these maps are continuous covering maps, use as a basis of the topology the sets $U_\alpha = \left\{ r_{U,y}(\alpha )\right\}$ , where $U$ is the interior of an embedded disk in a Euclidean neighborhood and $\alpha \in H_n^{M,U}$ is a fixed element, with $r_{U,y}$ induced by inclusion.

Problem 6.15.13

(a)(b)

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(a)

Show that nearby points in $\tilde{M}$ (notation as in Exercise 6.15.12) have the same divisibility as elements of groups isomorphic to $\mathbb {Z}$ .

(b)

Show that $\tilde{M}$ consists of the disjoint union of one copy of the identity covering space (corresponding to the $0$ element of $H_n^{M,x}$ for each $x$ ) and a natural number of copies of covering spaces equivalent to $(\hat{M}, \hat{p}, M)$ , one for each divisibility $d \in \mathbb {N}$ .

Problem 6.15.14

(a)(b)(c)

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(a)

Show that the homology-orientability condition on $M$ (Definition 6.15.1) is equivalent to finding a continuous map $s : M \to \hat{M}$ so that $\hat{p}s(x) = x$ (notation as in Exercise 6.15.12). Such a map $s$ is called a section of the covering space.

(b)

Show that if there is a nonzero section of $p : \tilde{M} \to M$ (so that the image of $x$ is not the zero element of $H_n^{M,x}$ ), then there is a section of $\hat{M}$ .

(c)

Show that $M$ is nonorientable iff the only section of $p : \tilde{M} \to M$ is the zero section.

Problem 6.15.15

Write $\Gamma_M$ for the sections of the covering space $\hat{M}$ (Exercise 6.15.12, Exercise 6.15.14). Show that if $s_1, s_2 \in \Gamma_M$ and $s_1(x) = s_2(x)$ for one $x$ , then $s_1 = s_2$ .

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Problem 6.15.16

Show that there exists a section $s : M \to \hat{M}$ (Exercise 6.15.14) iff the covering space $(\hat{M}, \hat{p}, M)$ is equivalent to two copies of the identity covering space, or, equivalently, $\hat{M}$ is the union of two components, each of which is sent homeomorphically to $M$ via $\hat{p}$ .

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Problem 6.15.17

Show that $M$ is orientable (Definition 6.15.1) iff $\hat{M}$ is homeomorphic to the disjoint union of two copies of $M$ .

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Problem 6.15.18

Use the theory of covering spaces to show that a simply connected manifold is orientable.

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Remark.

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Nonorientability leads to a nontrivial connected double covering.

Problem 6.15.19

(a)(b)

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(a)

Show that if $\pi_1(M, x)$ does not have an index- $2$ subgroup, then $M$ is orientable.

(b)

Give an example of a surface which is orientable and whose fundamental group contains an index- $2$ subgroup.

Problem 6.15.20

Suppose that $M$ is connected. Show that $\Gamma_M$ (the sections of $\hat{M}$ , Exercise 6.15.15) has the structure of a group, and the evaluation map $E : \Gamma_M \to H_n^{M,x} \simeq \mathbb {Z}$ given by $E(s) = s(x)$ is an injective homomorphism (monomorphism) for all $x \in M$ .

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Problem 6.15.21

(a)(b)

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(a)

Show that if $M$ is connected and orientable, then $\Gamma_M \simeq \mathbb {Z}$ , with isomorphism given by $E$ (Exercise 6.15.20).

(b)

Show that if $M$ is connected and nonorientable, then $\Gamma_M = 0$ .

Problem 6.15.22

(a)(b)

For a manifold $M$ with possibly empty boundary, there is a map $S_M : H_n(M, \partial M) \to \Gamma_{\mathrm{int}\, M}$ so that $S_M(\alpha )(x) = r_x(\alpha )$ .

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(a)

Show that $S_M(\alpha )$ satisfies the local consistency condition required for a section.

(b)

Show that $S_M$ is a homomorphism.

Problem 6.15.23

(a)(b)(c)

Suppose that $M$ is connected and compact, $\partial M = \emptyset$ .

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(a)

$H_i(M) = 0$ , $i > n$ .

(b)

If $M$ is nonorientable, then $H_n(M) = 0$ .

(c)

If $M$ is orientable via a section $s(x) = \mu_x$ , then there exists $\mu \in H_n(M)$ with $r_x(\mu ) = \mu_x$ .

Remark.

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For part (c), take $A = M$ in Theorem 6.15.3 (the main tool for this exercise and Exercises 6.15.24-6.15.35): let $M$ be a connected $n$ -manifold, with $\partial M = \emptyset$ , and let $A$ be a compact subset of $M$ . (1) $H_i^{M,A} = 0$ , $i > n$ . (2) If $\alpha \in H_n^{M,A}$ and $r_{A,x}(\alpha ) = 0$ for all $x \in A$ , then $\alpha = 0$ . (3) If $M$ is oriented via a section $s(x) = \mu_x$ , then there exists $\mu_A \in H_n^{M,A}$ with $r_{A,x}(\mu_A) = \mu_x$ . Proposition 6.15.1 (remark to Exercise 6.15.11) asserts that Theorem 6.15.3 is true for $M = \mathbb {R}^{n}$ and $A$ a compact convex set.

Problem 6.15.24

Suppose $M$ is connected and is not compact and $\partial M = \emptyset$ . Let $i \geq n$ . Let $\alpha \in H_i(M)$ be represented by a chain $c$ and suppose $S$ is the support of $c$ .

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(a)

Show that there is an open set $U$ with $S \subset U$ and $\bar{U}$ compact.

(b)

Let $V = M \setminus \bar{U}$ . Show that $H_i(U) \simeq H_i(U \cup V, V)$ .

(c)

Use the long exact sequence of the triple $(M, U \cup V, U)$ and Theorem 6.15.3(1) (Exercise 6.15.23) to show that $H_i(U \cup V, V) = 0$ for $i > n$ .

(d)

Show that $\alpha = 0$ when $i > n$ .

(e)

For the remaining parts, let $i = n$ . Show that $r_x(\alpha ) = 0$ for all $x \in M$ by evaluating it at $x \in V$ .

(f)

By first regarding $c$ as a chain in $(U \cup V, V)$ and using the long exact sequence of the triple, show that $\alpha = 0$ .

Problem 6.15.25

(a)(b)(c)(d)

Exercises 6.15.23--6.15.24 have taken care of Theorem 6.15.2(1)--(3) (stated in the remark to Exercise 6.15.6) when $\partial M = \emptyset$ . The following exercises assume $\partial M \neq \emptyset$ , using the fact that there is a collar neighborhood $C$ of $\partial M$ homeomorphic to $\partial M \times [0,1]$ .

Let $C \simeq_h \partial M \times [0,1]$ be a collar. Let $C_t = h[0,t)$ and $A_t = M \setminus C_t$ , $t \leq 1$ .

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(a)

Show that $M$ , $\mathrm{int}\, M$ , and $A_t$ deformation-retract to $A_1$ .

(b)

Show that $(M, C_t)$ deformation-retracts to $(M, \partial M)$ for $t < 1$ .

(c)

Show that $H_i(M) \simeq H_i(\mathrm{int}\, M)$ .

(d)

Show that the map induced by inclusion $H_i(M, \partial M) \to H_i^{M, A_t}$ is an isomorphism for all $t$ .

Problem 6.15.26

Using Exercise 6.15.25 (for $M$ with $\partial M \neq \emptyset$ ), show that $H_i(M) = 0$ , $i \geq n$ (taking care of the remaining cases of Theorem 6.15.2(1)--(4)).

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Problem 6.15.27

Suppose $M$ is compact and orientable with $\partial M \neq \emptyset$ . Let $s(x) = \mu_x \in \Gamma_{\mathrm{int}\, M}$ be a section for $x \in \mathrm{int}\, M$ . Show that there exists a class in $H_n(M, \partial M)$ which maps to $\mu_x$ under the map $r_x : H_n(M, \partial M) \to H_n^{M,x}$ for each $x \in \mathrm{int}\, M$ (this takes care of Theorem 6.15.2(5)).

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Problem 6.15.28

We now return to prove Theorem 6.15.3 (stated in the remark to Exercise 6.15.23). The key is to examine the relative Mayer--Vietoris sequence connecting the homology of $(M, M \setminus C)$ when $C$ takes on the values $C = A, B, A \cap B, A \cup B$ , using $(M \setminus A) \cup (M \setminus B) = M \setminus (A \cap B)$ , $(M \setminus A) \cap (M \setminus B) = M \setminus (A \cup B)$ .

Show that if $U, V$ are open sets in $M$ , then there is a long exact sequence with segment

H_{k+1}(M, U \cup V) \xrightarrow {\delta } H_k(M, U \cap V) \xrightarrow {i_k} H_k(M,U) \oplus H_{k+1}(M,V) \xrightarrow {j_k} H_k(M, U \cup V).

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Remark.

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Start with the diagram

\begin{array}{ccccccccc}& 0 & & 0 & & 0 & & & \\ & \downarrow & & \downarrow & & \downarrow & & & \\ 0 \to & S_k(U \cap V) & \to & S_k(U) \oplus S_k(V) & \to & S_k(U) + S_k(V) & \to & 0 \\ & \downarrow & & \downarrow & & \downarrow & & & \\ 0 \to & S_k(M) & \to & S_k(M) \oplus S_k(M) & \to & S_k(M) & \to & 0 \\ & \downarrow & & \downarrow & & \downarrow & & & \\ 0 \to & S_k(M, U \cap V) & \to & S_k(M,U) \oplus S_k(M,V) & \to & S_k(M)/(S_k(U)+S_k(V)) & \to & 0 \\ & \downarrow & & \downarrow & & \downarrow & & & \\ & 0 & & 0 & & 0 & & & \end{array}

and consider the horizontal rows as chain complexes and the vertical columns as giving an exact sequence of chain complexes. Look at the long exact sequence in homology to see that it says that the bottom horizontal row is exact. Then take the long exact sequence arising from this exact row. Then show that the fact that there is an isomorphism induced by inclusion between $H_k^{\left\{ U,V\right\} }(U \cup V)$ and $H_k(U \cup V)$ leads to an isomorphism between $H_k(M, U \cup V)$ and $H_k(S(M)/(S(U) + S(V)))$ .

Problem 6.15.29

Use Exercise 6.15.28 to show that if $A, B$ are closed subsets of $M$ , then there is a long exact sequence

\cdots \to H_{k+1}^{M, A \cap B} \to H_k^{M, A \cup B} \xrightarrow {i_k} H_k^{M,A} \oplus H_k^{M,B} \xrightarrow {j_k} H_k^{M, A \cap B} \xrightarrow {\delta } \cdots .

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Problem 6.15.30

The exact sequence of Exercise 6.15.29 is the basis for most of the remaining arguments proving Theorem 6.15.3 (Exercise 6.15.23): show that if $A, B, A \cap B$ satisfy the conclusion of Theorem 6.15.3, then so does $A \cup B$ .

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Problem 6.15.31

The point of Exercise 6.15.30 is that we can inductively build up sets $C$ satisfying the conclusion of Theorem 6.15.3 (Exercise 6.15.23) by starting with pieces $A, B, A \cap B$ having the property and using $C = A \cup B$ . If $C$ is a compact subset of a coordinate neighborhood $U \subset M$ homeomorphic to $\mathbb {R}^{n}$ , then $H_k^{M,C} \simeq H_k^{\mathbb {R}^{n},C}$ by excision, so it suffices to prove these statements when $M = \mathbb {R}^{n}$ . The starting point is when $M = \mathbb {R}^{n}$ and $C$ is a cube of some dimension $k$ with $0 \leq k \leq n$ (a product of intervals $I_1 \times \cdots \times I_n$ , each interval of the form $[a_i, b_i]$ with $a_i \leq b_i$ ) -- this case is Proposition 6.15.1 (remark to Exercise 6.15.11).

Now consider the space $C^s$ formed from an $n$ -cube by subdividing it using subdivisions of each subinterval. This is built up from subcubes by adding them one step at a time, where the intersections are finite unions of cubes of lower dimension; think of this as a CW complex whose cells are embedded cubes of various dimensions. Let $K$ be any subcomplex of $C^s$ .

Show that if $K$ is a subcomplex of $C^s$ , then the conclusion of Theorem 6.15.3 holds for $K$ .

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Remark.

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Use induction on the number of cells in $K$ and Exercise 6.15.30.

Problem 6.15.32

(a)(b)(c)(d)

Suppose $A$ is a compact subset of $\mathbb {R}^{n}$ , contained in a large cube $C$ . By subdividing this cube into subcubes successively (say subdivide the edges in half, fourths, etc.), let $K^i$ be the subcomplex of the $i$ th subdivision consisting of all closed subcubes which intersect $A$ somewhere in their interior (notation as in Exercise 6.15.31). We have $\cdots \subset K^2 \subset K^1 \subset K^0 = C$ and $A \subset K^i$ for all $i$ , with $A = \bigcap_i K^i$ .

Suppose $\alpha \in H_i^{\mathbb {R}^{n}, A}$ is represented by a chain $c$ . Look at $\partial c$ : it is a finite sum of singular simplices, each with image in $\mathbb {R}^{n} \setminus A$ . Let $B \subset \mathbb {R}^{n} \setminus A$ be the compact set consisting of the images of these simplices.

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(a)

Show that there is a minimal distance between points of $A$ and points of $B$ . Use this to show that there is an integer $p$ so that $c$ also represents a cycle of $S_i(\mathbb {R}^{n}, \mathbb {R}^{n} \setminus K^p)$ .

(b)

Show that (a) implies that $\alpha = 0$ when $i > n$ , since it is in the image of $H_i(\mathbb {R}^{n}, \mathbb {R}^{n} \setminus K^p)$ .

(c)

Suppose $i = n$ and $r_{A,x}(\alpha ) = 0$ for $x \in A$ . Show that this means that if $\alpha_K$ is the class represented by $c$ , then $r_{K^p, x}(\alpha_K) = 0$ for $x \in K^p$ . Then show that this implies that $\alpha_K = 0$ and hence $\alpha = 0$ .

(d)

Show that the usual orientation of $\mathbb {R}^{n}$ (Proposition 6.15.1) provides the class $\mu_A$ that restricts to the orientation class $\mu_x$ for $x \in A$ in the statement of Theorem 6.15.3.

Problem 6.15.33

We have now proved Theorem 6.15.3 (Exercise 6.15.23) when $M = \mathbb {R}^{n}$ and $A$ is any compact subset (Exercise 6.15.32). Show that Theorem 6.15.3 holds for a general connected $n$ -manifold $M$ when $A$ is a compact subset of a coordinate neighborhood $U$ .

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Problem 6.15.34

Show that any compact set $A \subset M$ can be written as the union $A = A_1 \cup \cdots \cup A_m$ of a finite number of sets $A_i$ which are compact sets in a Euclidean neighborhood $U_i \subset M$ .

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Problem 6.15.35

By using induction on the number of sets $m$ in the description of Exercise 6.15.34, and Exercise 6.15.33, complete the proof of Theorem 6.15.3 (Exercise 6.15.23).

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