Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

So far, any source I consult will gladly talk about cobordism classes of closed (compact and without boundary) oriented manifolds, but I have yet to see an example of a pair of manifolds which are not cobordant.

As far as I can tell, every pair of one- or two-dimensional closed oriented manifolds are bordant, so such an example would seemingly only occur in dimensions 3 or higher. They reason I think this is that since the torus and the sphere are bordant, we can just "iterate" this cobordism in a suitable way and exhaust all possibilities by the classification theorem of compact oriented surfaces.

I was wondering if anyone knows of an example of a non-cobordant pair of closed oriented manifolds which is relatively easy to appreciate. If possible, an excplicit construction would be preferrable.

share|improve this question

1 Answer 1

up vote 3 down vote accepted

You can just consider some simple bordism variants.

If you're considering oriented bordism, then the first example appears when considering bordisms between $4$-manifolds. The signature $\sigma(X^{4k})$ of an oriented $4k$-manifold $X^{4k}$ is an oriented bordism invariant. Now $$\sigma(S^4) = 0$$ and $$\sigma(\Bbb C P^2) = 1,$$ so $S^4$ and $\Bbb C P^2$ are not oriented bordant. In fact, $\Omega_4^{\mathrm{SO}} \cong \Bbb Z$ with generator $[\Bbb C P^2]$.

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.