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Betti Numbers for sphere and torus, cycles and Poincare Duality

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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.

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

  2. 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?

  3. 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?

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

  5. 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
asked Dec 25, 2014 in Theoretical Physics by leastaction (425 points) [ no revision ]
This question is too broad. If you don't know what a cycle is, you should read a bit of homology.

This post imported from StackExchange Physics at 2014-12-25 20:33 (UTC), posted by SE-user jinawee
I know that arbitrary linear combinations of submanifolds of a dimension p are called p-chains, and a chain that that has no boundary is called closed, and further a closed chain that is a cycle satisfies $\delta z_p = 0$. I am not sure how this helps me answer the rest of my question/questions. I do not know a whole lot about homology. Perhaps you can enlighten me with with some references, which may be more useful for a physics student?

This post imported from StackExchange Physics at 2014-12-25 20:33 (UTC), posted by SE-user leastaction
According to en.wikipedia.org/wiki/Chain_%28algebraic_topology%29 "A 0-cycle is a linear combination of points such that the sum of all the coefficients is 0. Thus, the 0-homology group measures the number of path connected components of the space." Okay so this answers what a cycle is, a 0-cycle is, etc. I'll check out Hatcher's book on Algebraic Topology. In the meantime, can you (or an OP) answer the rest of the questions?

This post imported from StackExchange Physics at 2014-12-25 20:33 (UTC), posted by SE-user leastaction
There are experts here who know way more than I do, but I think Nakahara is a good concise introduction to this topics. A p-cycle is just a p-chain with no boundary, $\partial_p c=0$.

This post imported from StackExchange Physics at 2014-12-25 20:33 (UTC), posted by SE-user jinawee
Don't rely on Becker, Becker and Schwarz for these topics; the introduction to Poincaré duality, cycles, and so forth is extremely brief, meant to ensure the reader can follow the mathematics of the text only.

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

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