Dold-Thom's theorem

In the mathematical field of algebraic topology , Dold-Thom's theorem calculates the homotopy groups of symmetrical products . The realization of the homology groups of any CW-complex as homotopy groups allows a calculation of homology in different contexts by means of homotopy-theoretical methods and it enables the transfer of methods of algebraic topology into algebraic geometry , for example the homotopy theory of algebraic varieties and motivic cohomology developed by Wojewodski .

Dold-Thom's theorem

Let be a connected CW-complex and its symmetric product, i. H. the Kolimes the quotient of under the action of the symmetrical group . Then applies ${\displaystyle X}$ ${\displaystyle SP(X)}$ ${\displaystyle X^{n}}$ ${\displaystyle S_{n}}$

${\displaystyle \pi _{i}(SP(X))=H_{i}(X)}$

for everyone . ${\displaystyle i\geq 1}$

literature

• Albrecht Dold , René Thom : Quasi-fibers and infinite symmetrical products. Ann. of Math. (2) 67: 239-281 (1958).

Web links

• Barnet-Lamb: The Dold-Thom Theorem