Algebras of the equivariant cohomologies of an $\mathfrak F$-classifying $T^k$-spaces

I. V. Usimov

Chair of Geometry, Topology and Mathematics Teaching Principles, Belarus State University, 4 Nezavisimosti Ave., Minsk, 200050 Republic of Belarus

Abstract: We consider equivariant cohomologies generated by the Borel functor $ E_\mathfrak F$ for the family of orbit types $\mathfrak F\subset\mathrm{Conj}_G$, which translates equivariant homotopy category EQUIV-HOMOT in $\mathfrak F$-isovariant homotopy category $\mathrm{ISOV}_\mathfrak F$-$\mathrm{HOMOT}$. Due to the effect of concentration of isovariant absolute extensors $\mathrm{ISOV}_\mathfrak F$-$\mathrm{AE}$ we calculate in explicit form the algebra of equivariant cohomologies of an $\mathfrak F$-classifying $G$-spaces for finite families of orbit types $\mathfrak F\subset\mathrm{Conj}_G$ in the case of actions of $k$-dimensional torus $G=T^k$.

Keywords: equivariant cohomologies, classifying $G$-spaces, isovariant absolute extensor, universal Palais $G$-space.

UDC: 515.146

Received: 07.07.2013

 English version: Russian Mathematics (Izvestiya VUZ. Matematika), 2015, 59:1, 51–59

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