1
$\begingroup$

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?

1 Answers 1

2

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.

  • 0
    What is the definition of boundary of a "cobordims" not manifold itself?2012-02-28
  • 0
    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)$2012-02-28
  • 0
    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?2012-02-28
  • 1
    Could you tell me where exactly you found this sentence? I'm having trouble with the context.2012-02-28
  • 0
    Please take a look at the proof of Theorem 4.3 on page 189 of this:http://books.google.com/books?id=w7dActmezxQC&q=computation+of+anomalies#v=snippet&q=computation%20of%20anomalies&f=false2012-02-28
  • 0
    Yes, looking at it carefully, I believe you are right. They state that $\partial W$ is formed by gluing the handlebodies $U$ and $V$ to $M$, using the identifications $\partial U\rightarrow \partial_-M$ and $\partial V\rightarrow \partial_+M$. Then they say "The manifold triple $(W,U,V)$ is a 4-dimensional cobordism with 'boundary' $(M,\partial_-M,\partial_+M )$." By the "boundary of a cobordism" they must means $\partial(W,U,V)=(\partial W,\partial U,\partial V)$. I haven't seen this definition in the literature, but maybe I need to read more2012-02-28
  • 0
    The idea of "manifold with corners" might play a role here? This is how you end up in situations where $\partial \partial W \not= \emptyset$.2012-11-01