In the book *String Theory and M-Theory* by Becker, Becker and Schwarz, the Betti Number $b_p$ is defined as the number of $p$-cycles which are not boundaries.

What is a $p$ cycle? Does it have to be simply connected?

For a sphere, $b_1 = 0$ if I infer the 1-cycle to be any simply connected loop on the surface of the sphere $S_2$, since it every such loop is contractible to a point. What are the corresponding interpretations for $b_0$ and $b_2$? Is a $0$-cycle a point?

According to the book, to every closed $p$-form $A$ there corresponds a $(d-p)$-cycle $N$ with the property

$\int_M A \wedge B = \int_N B$

for all closed $(d-p)$-forms $B$. How does one prove this identity?

How does one use Poincare duality to determine the Betti numbers of a manifold?

For $S^N$, do the Betti numbers alternate between 0 and 1? What does this signify?

This post imported from StackExchange Physics at 2014-12-25 20:33 (UTC), posted by SE-user leastaction