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Let X be a topological space and $C_n(X)$ be the singular chain complex. The homology is defined to be $H_n(X)$ = $ ker \partial_n / im \partial_{n+1}$.

What happens if we take $ K_n(X) = C_n(X) / im \partial_{n+1}$ instead?

(The idea comes from comparing the fundamental group to the fundamental groupoid)

  • 0
    Have you tried to compute such thing, for example? Try and see for yourself what happens.2012-11-06
  • 4
    I don't understand the downvote. The question is well-formulated and it's an interesting idea which is motivated by analogy with another, better-known interesting idea. It's certainly not a homework problem.2012-11-06
  • 2
    It's not invariant under homotopy.2012-11-06
  • 0
    Neither is the fundamental groupoid.2012-11-07

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