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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?
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