[Paper Review] Some geometric equivariant cohomology theories

Haggai Tene|arXiv (Cornell University)|Oct 30, 2012

Homotopy and Cohomology in Algebraic Topology20 references3 citations

TL;DR

This paper introduces a geometric construction of equivariant cohomology theories using stratifolds, establishing Poincaré duality via a new theory called backwards (co)homology. It defines stratifold Tate (co)homology as a relative theory unifying these, and provides explicit computations of cup products in the cohomology of LBG via geometric representatives, particularly for G=S^3.

ABSTRACT

In this paper we give a geometric construction of the Borel equivariant (co)homology for spaces with a $G$-action, where $G$ is a compact Lie group with the property that the adjoint representation is orientable. A nice feature of these constructions is that there are corresponding Poincaré dual (co)homology theories called backwards (co)homology. This gives rise to a third relative (co)homology theory which we call stratifold Tate (co)homology. These Tate groups agree with the original definition of Tate cohomology for finite groups given by Swan. All constructions in this paper are geometric and use stratifolds. One advantage of this description is that elements in these groups can be described concretely by representatives. We give some examples of that.

Motivation & Objective

• To develop a geometric, bordism-based construction of equivariant (co)homology theories for compact Lie groups with orientable adjoint representation.
• To introduce backwards (co)homology as a Poincaré dual to equivariant stratifold (co)homology.
• To define stratifold Tate (co)homology as a relative theory unifying the two via natural transformations.
• To provide explicit geometric representatives for cohomology classes and compute products in equivariant cohomology, especially for LBG.
• To demonstrate the utility of the geometric approach by computing the product structure in H*(LBG) for G=S^3.

Proposed method

• Construct equivariant stratifold homology and cohomology using bordism classes of equivariant maps from oriented G-stratifolds to G-spaces.
• Define backwards (co)homology as the Poincaré dual theory via duality between compact stratifolds (homology) and proper Fredholm maps (cohomology).
• Use the natural transformation between backwards cohomology and equivariant stratifold cohomology to define a relative theory: stratifold Tate (co)homology.
• Leverage the homotopy equivalence G^{ad} ×_G EG ≃ LBG to identify H*(LBG) with H*_G(G^{ad}) and transfer geometric structures.
• Apply Poincaré duality in the stratifold framework to construct the cup product in cohomology via composition of representatives in backwards homology.
• Use the group multiplication μ: G^{ad} × G^{ad} → G^{ad} to induce the product on backwards homology, then dualize to obtain the cup product in cohomology.

Experimental results

Research questions

• RQ1 How can equivariant (co)homology theories be constructed geometrically using stratifolds, rather than singular homology?
• RQ2 What is the geometric meaning of Poincaré duality in the equivariant setting, and how can it be realized via a dual theory?
• RQ3 How can the Tate (co)homology group be defined geometrically as a relative theory combining equivariant stratifold and backwards (co)homology?
• RQ4 What is the explicit geometric description of the cup product in H*(LBG) for G=S^3, and how does it relate to the product in equivariant cohomology?
• RQ5 Can the geometric approach simplify known constructions, such as the product in LBG cohomology defined via Umkehr maps?

Key findings

• The paper establishes a natural isomorphism between equivariant stratifold homology SH^G_k(X) and the Borel equivariant homology H_{k−dim(G)}^G(X;Z) for G-CW complexes.
• For G-Hilbert manifolds, equivariant stratifold cohomology SH^*_G(P) is naturally isomorphic to Borel equivariant cohomology H^*_G(P;Z).
• The backwards (co)homology theory DSH^*_G(M) is Poincaré dual to equivariant stratifold cohomology, providing a geometric realization of polar counterparts.
• The stratifold Tate (co)homology is defined via long exact sequences involving natural transformations between backwards and stratifold (co)homology.
• For G=S^3, the product * on H*(LBG) is explicitly computed: a_{4k}*a_{4l}=0, a_{4k}*b_{4l+3}=a_{4k+4l}, b_{4k+3}*b_{4l+3}=b_{4k+4l+3}, up to sign.
• The product in cohomology is realized geometrically as composition of representatives: α*β is represented by S×S′ → G^{ad}×EG×G^{ad}×EG → G^{ad}×EG via μ.