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Dold-Thom theorem

Theorem

(Dold-Thom)

The ordinary homology, hence the standard nn-th homology group of a CW-complex XX is isomorphic to the nn-th homotopy group of the free topological commutative monoid on XX, which is an infinite symmetric product: that is, the colimit Sym X\mathrm{Sym}^\infty X (denoted also SP SP^\infty) of the symmetric powers Sym NX=X*X*...*X=(X×X×...×X)/Σ N\mathrm{Sym}^N X = X*X*...*X = (X\times X\times ...\times X)/\Sigma_N of XX:

H i(X)=π i(colim NSym NX). H_i(X) = \pi_i (\mathrm{colim}_N\, \mathrm{Sym}^N X) \,.

The Mayer-Vietoris sequence for homology is a consequence of applying π *()\pi_*(-) to the homotopy pullback square resulting from the application of Sym \mathrm{Sym}^\infty to the homotopy pushout square formed by the inclusions of the intersection, ABA \cap B, of two subspaces AA and BB of a space XX into AA and BB.

References

The original article is

  • A. Dold, R. Thom, Quasifaserungen und unendliche symmetrische Produkte , Ann. Math. (2) 69 (1959), 239–281. (pdf)
Revised on October 19, 2014 10:29:09 by Thomas Holder (89.15.239.235)