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May 8, 2007

Cohomology and Computation (Week 22)

Posted by John Baez

This week in our seminar on Cohomology and Computation, we explained how homology ‘counts holes’:

  • Week 22 (May 3) - Cohomology and chain complexes. The functor from simplicial sets to simplicial abelian groups. The functor from simplicial abelian groups to chain complexes. The homology of a chain complex as a general method of ‘counting holes’. Some examples: the hollow triangle has H 1=H_1 = \mathbb{Z} because it has a ‘1-dimensional hole’. The twice filled triangle (a triangulated 2-sphere) has H 1={0}H_1 = \{0\} but H 2=H_2 = \mathbb{Z} because it has a ‘2-dimensional hole’.

Last week’s notes are here; next week’s notes are here.

Next time we’ll start describing how to ‘count holes’ in any algebraic gadget described using a monad!

Posted at May 8, 2007 1:49 AM UTC

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Read the post Cohomology and Computation (Week 21)
Weblog: The n-Category Café
Excerpt: Why mathematicians like to take algebraic gadgets and topological spaces and turn them into simplicial sets.
Tracked: June 18, 2007 7:32 AM

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