From: Arthur Wasserman <awass@math.lsa.umich.edu>
Newsgroups: sci.math.research
Subject: Re: RP^n as a boundary of a manifold
Date: Sat, 29 Apr 95 10:55:49 -0500

In article <3njrcm$qr4@pith.uoregon.edu> Jeffrey Boersema,
euclid!boersema@uunet.uu.net writes:
>I am interested in seeing an explicit construction of a manifold
>with boundary RP^n

If n = 2k-1 then RP^n is the quotient of the unit sphere in R^(n+1) = R^2k = C^k
by Z mod2. But C^k admits an S1 action and this action gives a free action of
S1/Z mod2 on RP^n. The manifold you want is the quotient of RP^n x D^2 by S1/Z2.
D^2 is the unit disk in C^1 with the action by S1/Z mod 2 that is free off the
origin.

[Mod. note:  I misread this the first time.  Another way of putting
it is that RP^n is a circle bundle over CP^(k-1), and so you can
take the disk bundle with the same gluing maps. - Greg]