nLab
Dold-Thom theorem

Theorem

(Dold-Thom)

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 ${\mathrm{Sym}}^{\mathrm{\infty }}X$ (denoted also ${\mathrm{SP}}^{\mathrm{\infty }}$) of the symmetric powers ${\mathrm{Sym}}^{N}X=X*X*...*X=(X\times X\times ...\times X)/{\Sigma }_N$ of $X$:

$$H_i(X)={\pi }_i({\mathrm{colim}}_N{\mathrm{Sym}}^{N}X).$$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 ${\mathrm{Sym}}^{\mathrm{\infty }}$ 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.