TITLE: Is there an explicit description of a cobordism between $\mathbb{CP}^n$ and $\mathbb{RP}^n\times\mathbb{RP}^n$?
QUESTION [29 upvotes]: 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).

REPLY [46 votes]: 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: