Excuse the very basic question:

I'm following Milnor's Lectures on Characteristic Classes. He defines a relation on the collection of compact, smooth, oriented manifolds of dimension $n$ by letting $M_1\sim M_2$ if $M_1\sqcup -M_2$ is a boundary (If $M$ is a manifold, then $-M$ is the same manifold with the opposite orientation). He uses this to define cobordism classes and further goes on to define a graded ring structure on these classes. He says clearly the relation is reflexive and symmetric. I see symmetric, but I'm not entirely sure why this relation is reflexive. Since we're supposed to have a group structure, and the $0$ element is the empty set, I'm assuming that the boundary of $M\sqcup -M$ is the empty set. Why is it that $\partial (M\sqcup -M)=\emptyset $ ? Do the two manifolds somehow cancel each other out since they have opposite orientations? How can I picture this?


You're a bit confused about what this definition means. First of all, the $n$-manifolds we're considering here all do not have boundary. So $M$ should be an $n$-manifold without boundary, and you shouldn't be thinking about its boundary at all.

The relation $M_1\sim M_2$ means that $M_1\sqcup -M_2$ is the boundary of some $(n+1)$-manifold with boundary. So to prove $M\sim -M$, you want to prove that $M\sqcup -M$ is the boundary of some $(n+1)$-manifold. To prove this, you can just take the $(n+1)$-manifold to be $N=M\times[0,1]$. The boundary of $N$ is $M\times \{0\}\cup M\times\{1\}$, where $M\times\{1\}$ has the given orientation of $M$ and $M\times\{0\}$ has the opposite orientation, so $\partial N\cong M\sqcup -M$ as an oriented manifold.

  • $\begingroup$ Yes, you are quite right, I was very confused! Thank you for your help! $\endgroup$ Dec 5 '17 at 4:38

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