Cohomology of toric diagrams

par M. Grigory Solomadin (University of Strasbourg)

mardi 2 avr. 2024, 14:00 → 15:00 Europe/Paris

IMT 1R2 207 (Salle Pellos)

For a topological group G, a nice class of G-spaces is given by those G-CW spaces that are obtained by attaching finitely many equivariant cells with strictly increasing 

G-stabilizer subgroups.

Another way of describing such spaces is in terms of homotopy colimits of G-orbit diagrams in the category of compactly generated topological spaces.

By restricting to the case of a compact torus G, one obtains the class of toric diagrams consisting of quotient tori and homomorphisms between them, indexed by a finite category.

Toric diagrams encode (up to a T-equivariant homotopy) any toric variety, quotients of moment-angle complexes and other interesting spaces studied in toric geometry and toric topology.

In the talk, new formulas for singular cohomology groups of such spaces (for G=T and rational coefficients) will be presented in terms of sheaf cohomology groups over Alexandrov spaces.

These groups are known by the name of cohomology for the respective (toric) diagram, and appear at the second page of the Bousfield-Kan cohomological spectral sequence.

The spectral sequence in question converges to the cohomology of the homotopy colimit of the diagram.

Our main tool is the collapse of this spectral sequence at its second page, implied by formality of any toric diagram.

If time permits, a possible approach to integral coefficients and relation to open conjectures from toric geometry will be presented.

The talk is based on arXiv:2401.14146.