nLab Dold-Thom theorem

Definition

(Symmetric Product)

There is a natural forgetful functor from topological commutative monoids to pointed topological spaces, where the basepoint of a commutative monoid is taken to be its unit element. The induced monad is the infinite symmetric product construction:

$$Sym(X,e) = \mathrm{colim}_N Sym^N(X) \qquad Sym^N(X) = (X \times \dots \times X)/\Sigma_N$$

Here the colimit is taken over the maps $Sym^N(X) \to Sym^{N+1}(X)$, $(x_1,\dots, x_n) \mapsto (x_1,\dots, x_n, e)$ where $e$ is the basepoint.

Thus $Sym(X,e)$ is the free topological commutative monoid on $X$ with the basepoint $e$ acting as the unit.

Theorem

(Dold-Thom)

The ordinary homology, hence the standard $n$-th homology group of a pointed connected CW-complex? $(X,e)$ is isomorphic to the $n$-th homotopy group of the infinite symmetric product of $(X,e)$

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

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)

For a textbook account, see Hatcher, Sec. 4.K p. 475.