Definition of a 4-cobordism with boundary

Question

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?

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.