What does $P\times_G V\to B$ mean?

Question

Let $$\pi:P\to B$$ be a principal $G$-bundle and $$\rho:G\times V\to V$$ a continuous action of $G$ on the vector space $V$.

What does the notation $P\times_G V\to B$ mean?

It is supposed to be a vector bundle over $B$, but I am not sure how. I've never seen this notation before. Is there a textbook explaining this concept somewhere? How is it called?

Answer 1

This is the associated vector bundle. It is constructed from the sum bundle $P\oplus V$ by quotienting by the action of $G$. The action is simply to act on both components simultaneously, i.e., $g.(p,v)=(g.p, g.v)$. A good source for this sort of thing is the book by Lawson and Michelsohn:

Lawson, H. Blaine, Jr.; Michelsohn, Marie-Louise. Spin geometry. Princeton Mathematical Series, 38. Princeton University Press, Princeton, NJ, 1989.

See http://www.ams.org/mathscinet-getitem?mr=1031992