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Show that if $X$ may be deformed into $Z$ then $X$ and $Z$ are cobordant.

Deformation definition: deformation of a submanifold $Z$ in $Y$ is a smooth homotopy $i_t:Z\to Y$ where $i_o$ is the inclusion map $Z\to Y$ and each $i_t$ is an embedding.

Cobordant definition : Two compact submanifolds $X$ and $Z$ in $Y$ are cobordant if there exist a compact manifold with boundary $W$ in $Y \times I$ such that $\partial W= X\times \{0\} \cup Z \times \{1\}$

Boundary Theorem: suppose that $X$ is the boundary of some manifold $W$ and $g: X \to Y$ is a smooth map. If $g$ may be extended to all of $W$ then $I_2 (g,Z)=0$ for any closed submanifold $Z$ in $Y$ of complementary dimension.

Assume that $X$ deformed into $Z$ then there is a homotopy $i_t: X \to Z$ such that $i_0 (x)=x$ adn $i_t$ is embedding, meaning for each $T$, $i_t$ is injective and the preimmage of any compact set it also compact.

I can't see how these information help me to prove this statement.

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XiaoXiao Zhen
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1 Answers1

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This is where pictures really make the intuition clear. I visualize a smooth homotopy $i_t \colon X \to Z$ as a "curvy cylinder" where the bottom of the cylinder is $i_0 \cong X$ and the top of the cylinder is $i_1 \cong Z$. Note that I am using $\cong$ to denote topologically isomorphic.

The bottom of this "cylinder manifold" is then clearly $X \times \{0\}$ while the top is $Y \times \{1\}$. So the boundary of this "cylinder manifold" is just the union of the two.

The definitions you use are embedding $X$ and $Z$ in some ambient manifold $Y$, so a "curvy cylinder" between $X$ and $Z$ should be some "curvy cylinder submanifold" of $Y \times I$.

That's the intuition. For the (not so painful) rigor, see the spoilers below:

Let $Y$ be a manifold and $X,Z \subseteq Y$ be submanifolds. Also, $X$ be deformable into $Z$. That is, there exists a smooth homotopy $i \colon X \times [0,1] \to Y$ where $i_0 \equiv i(X,0) = X \times \{0\} \cong X$ and $i_1 \equiv i(X,1) = Z \times \{1\} \cong Z$.

Now define $g \colon X \times I \to Y \times I$ where $g(x,y) = i_y(x) \times \{x\}$. Since each $i_x$ is an embedding, $g$ is also an embedding $W \equiv g(X,Y) \subseteq Y \times I$. Note that via the obvious chart maps, $W$ has a boundary, namely $\partial W = X \times \{0\} \cup Y \times \{1\}$.

Furthermore, since by definition $X$ is a compact manifold, then via the embedding $g$ you can pull back any open covering of $W$ onto $X \times I$ to find a countable subcover, thus $W$ is compact.

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