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Definition of a 3-cobordism (in my context) is a pair $(M, \partial_{-} M, \partial_{+}M)$, where $M$ is a closed orientable topological 3-manifold and $\partial M$ is a disjoint union of $\partial_{-} M$ and $\partial_{+}M)$.

I have a question regarding the following sentence:

"$(W, U, V)$ is a 4-dimensional cobordism with boundary $(M, \partial_{-} M, \partial_{+}M)$."

Here $W$ is a 4-dimensional manifold and $U, V$ are 3 dimensional manifolds.

  1. What is the definition of 4-dimensional cobordism used here? Is it just same as the 3-dimensional case? That is, $\partial W$ is a disjoint union of $U$ and $V$.

  2. Also what is the definition of the term boundary used above?

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1 Answer 1

An $n$-dimensional manifold $M$ with boundary is a topological space where every point $p\in M$ has a nhd $U$ which is homeomorphic (via some homeomorphism $\phi$) to an open subset of the upper-half plane $ \mathbb{R}^n_{+} = \{ (x_1,\dots,x_n)\in \mathbb{R}^n\ |\ x_1\geq0 \} $. The pair $(U,\phi)$ is called a "chart at $p$".

The boundary $\partial M$ is all of the points $p$ who have a chart $(U,\phi)$ where $\phi(p)$ is on the boundary of $\mathbb{R}^n_+$, i.e. it is of the form $\phi(p)=(0,x_2,\dots,x_n)$.

In general, given two compact $n$-dimensional manifolds $M,N$, a cobordism is a compact $(n+1)$-dimensional manifold $W$ with boundary such that $\partial W\cong M\dot{\cup}N$. Many people require their manifolds to be oriented, in which case $M$ or $N$ has to appear with reversed orientation.

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What is the definition of boundary of a "cobordims" not manifold itself? –  Link S Feb 28 '12 at 3:11
    
I'm not sure I understand what you're asking. A cobordism IS a manifold, with a very particular boundary. I think the confusion comes from the awkward phrasing of the sentence you have in quotes. I parse it in two ways: either they mean $\partial W= M,\partial U=\partial_- M$ and $\partial V=\partial_+{M}$, or they mean that $(W,U,V)=(M,\partial_- M,\partial_+ M)$ –  you Feb 28 '12 at 3:32
    
My guess is the first one you mentioned since $W$ is 4-manifold and $M$ is a 3-manifold. But in that case $\partial \partial W $ is not $0$ in general. Is it ok? –  Link S Feb 28 '12 at 3:45
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Could you tell me where exactly you found this sentence? I'm having trouble with the context. –  you Feb 28 '12 at 4:33
    
Please take a look at the proof of Theorem 4.3 on page 189 of this:books.google.com/… –  Link S Feb 28 '12 at 4:38

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