Fundamental class and top homology of manifolds

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 $S_M(\alpha )(x) = r_x(\alpha )$.

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

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 $c$.

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 $A_t = M \setminus C_t$, $t \leq 1$.

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)).

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)).

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

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

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 $A \cup B$.

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 $K$.

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.

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 $U$.

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 $U_i \subset M$.

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).