With a little bit of work, one can show that $\mathbb{CP}^n$ and $\mathbb{RP}^n\times\mathbb{RP}^n$ have the same Stiefel-Whitney numbers, so by a theorem of Thom, they are (unorientedly) cobordant.

Is there an explicit description of a cobordism between them?

I know the answer for $n = 1$: it reduces to finding a cobordism between $S^2$ and $S^1\times S^1$ (take a solid ball and remove a solid torus from the interior, or take a solid torus and remove a solid ball from the interior).


An explicit cobordism is given by Stong:

R. E. Stong, A Cobordism, Proceedings of the American Mathematical Society Vol. 35, No. 2 (Oct. 1972), pp. 584-586

I do like the short title "A Cobordism".

The construction is as follows:

Write $\mathbb{CP}^n$ with homogeneous coordinates $[z_0,z_1,\dots ,z_n]$ and consider the space $$W=(\mathbb{CP}^n\times [0,1])/\sim,$$ where $([z_0,z_1,\dots ,z_n],1)\sim([\bar{z_0},\bar{z_1},\dots ,\bar{z_n}],1)$ if $|z_0^2+z_1^2\dots +z_n^2|\leq\frac{3}{5}$. Then Stong takes a page to prove that $W$ is the desired cobordism.

But one should also read the note: enter image description here enter image description here


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

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