# nLab Dold-Thom theorem

###### Theorem

(Dold-Thom)

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

$H_i(X) = \pi_i (\mathrm{colim}_N\, \mathrm{Sym}^N X) \,.$

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

## 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)