Homotopy quotient

In mathematics , the homotopy quotient is a term from algebraic topology .

It allows a homotopy-invariant definition of the quotients of group effects and can therefore be used to define homotopy-invariants of group effects, for example the equivariant cohomology .

definition

A group works in one room . As homotopies quotient of this group is called the effect homotopy of , wherein a contractible space with a free -Wirkung is. ${\displaystyle G}$ ${\displaystyle X}$ ${\displaystyle X//G}$ ${\displaystyle (X\times EG)/G}$ ${\displaystyle EG}$ ${\displaystyle G}$

The type of homotopy of does not depend on the choice of the contractible, free space . For example, one can choose for the geometric realization of the simplicial complex , the -Simplices of which correspond to the tuples in . ${\displaystyle X//G}$ ${\displaystyle G}$ ${\displaystyle EG}$ ${\displaystyle EG}$ ${\displaystyle n}$ ${\displaystyle G^{n+1}}$

For free effects, homotopy is equivalent to , but in general it is not. ${\displaystyle X//G}$ ${\displaystyle X/G}$

Web links

• homotopy quotient (nLab)
• M. Anel: The homotopy quotient