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.

Say $X$ has the homotopy type of a CW-complex. The Dold-Thom theorem states that $\pi_i SP(X) \cong \tilde{H}_i(X;\mathbb{Z})$, where $SP(X)$ denotes the infinite symmetric product of $X$.

I am just curious about some useful applications of this theorem or instances where this theorem simplifies calculations significantly.

share|improve this question
add comment

1 Answer

For one you can get the Mayer-Vietoris sequence in homology almost immediately from a homotopy pushout, but depending on what angle you look at it from, that may not be that useful (as in, to prove the theorem, you need to define the functor $H_*$ anyway, and then you probably already know it satisfies M-V).

You can deduce a group structure on the Eilenberg-Mac Lane spaces, see this previous M.SE answer. (In fact it is probably possible to prove the existence of the Eilenberg-Mac Lane space using Dold-Thom)

At a philosophical level I can do no more than point you to Thomas Barnet-Lamb's excellent piece on Dold-Thom, and some of the reasons why it is important.

share|improve this answer
    
Thank your for the reference to Thomas Barnet-Lamb's paper. I will take a look at it in more depth tonight. I know about the direct applications/corollaries (For example those in Hatcher) of the theorem but I am more curious about its usefulness in computations or other places where it arises. –  confusedmath Mar 23 '12 at 21:40
add comment

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.