nLab Dold-Thom theorem


(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)=colim NSym N(X)Sym N(X)=(X××X)/Σ N 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)Sym N+1(X)Sym^N(X) \to Sym^{N+1}(X), (x 1,,x n)(x 1,,x n,e)(x_1,\dots, x_n) \mapsto (x_1,\dots, x_n, e) where ee is the basepoint.

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



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

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.


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.

Last revised on June 13, 2019 at 22:31:19. See the history of this page for a list of all contributions to it.