Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Is there any relation between the homology of a space with local coefficients (in $\mathbb Q$ vector space) and the homology with coefficients in $\mathbb Q$? Thanks!

share|improve this question
    
Is $Q$ the rational numbers? In general these two notions are the same if and only if the local coefficient system is un-twisted. See for example the local-coeffiecients section in Hatcher's Algebraic Topology text. –  Ryan Budney Oct 9 '10 at 18:05

1 Answer 1

If you compute homology with twisted coefficients, where the coefficient system involves vector spaces of dimension $d$, then the Euler characteristic of the resulting homology spaces (i.e. the alternating sum of the $H_i$ with twisted coefficients) is equal to $d$ times the Euler characteristic of the space computed via homology with trivial coefficients.

share|improve this answer
    
Do you know a reference for this fact on Euler characteristic? I'd love to add one to math.stackexchange.com/a/369762/274 ! –  Mariano Suárez-Alvarez Apr 22 '13 at 21:49
    
@MarianoSuárez-Alvarez: Dear Mariano, I'm not sure of a reference in general. I started to write down an argument here, but it was getting a bit long and meandering. I'll think a bit about how to make an efficient argument in at least some degree of generality, and if I succeed, add it to my answer. Cheers, –  Matt E Apr 23 '13 at 3:33

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.