[Paper Review] $RO(C_2)$-graded cohomology of equivariant Grassmannian manifolds

Eric Hogle|arXiv (Cornell University)|Jun 5, 2018

Geometry and complex manifolds5 references3 citations

TL;DR

This paper computes the $RO(C_2)$-graded Bredon cohomology of real and complex $C_2$-equivariant Grassmannians using an equivariant Schubert cell decomposition and a cellular spectral sequence. It provides explicit formulas for the cohomology of families such as $\operatorname{Gr}_k(\mathbb{R}^{n,1})$ and $\operatorname{Gr}_2(\mathbb{R}^{n,2})$, showing that these cohomology groups are free modules over the coefficient ring $\mathbb{M}_2 = H^{\bullet,\bullet}(\text{pt}; \underline{\mathbb{Z}/2})$, with weights determined by representation-theoretic data and nontrivial differentials in the spectral sequence.

ABSTRACT

We compute the $RO(C_2)$-graded Bredon cohomology of certain families of real and complex $C_2$-equivariant Grassmannians.

Motivation & Objective

• To compute the $RO(C_2)$-graded Bredon cohomology of real and complex $C_2$-equivariant Grassmannian manifolds.
• To extend known results on cohomology of equivariant Grassmannians beyond trivial cases by determining the $RO(C_2)$-grading, particularly the weights $b_i$ in the decomposition of cohomology as a free $\mathbb{M}_2$-module.
• To analyze the structure of the equivariant cellular spectral sequence arising from the Schubert cell filtration, identifying and computing nontrivial differentials that complicate the cohomology computation.
• To establish a duality between Schubert cells and their orthogonal complements via the perp map, using this to constrain the cohomology generators and their images under forgetful maps.

Proposed method

• Uses an equivariant Schubert cell decomposition to construct a cellular filtration on $\operatorname{Gr}_k(\mathbb{R}^{n,q})$ and $\operatorname{Gr}_k(\mathbb{C}^{n,q})$, enabling the use of a cellular spectral sequence.
• Applies the equivariant suspension isomorphism $\tilde{H}^{\bullet,\bullet}(S^{p,q} \wedge X) \cong \tilde{H}^{\bullet-p,\bullet-q}(X)$ to relate cohomology of suspensions to shifted cohomology of the base space.
• Employs the perp map $P: \operatorname{Gr}_k(\mathbb{F}^n) \to \operatorname{Gr}_{n-k}(\mathbb{F}^n)$, which sends Schubert cells $\Omega_\lambda$ to $\widehat{\Omega}_{\lambda^T}$, to relate cohomology of Grassmannians of complementary dimensions.
• Uses the fact that $H^{\bullet,\bullet}(X)$ is a free module over $\mathbb{M}_2 = H^{\bullet,\bullet}(\text{pt})$, with generators corresponding to cells, and computes the weights $b_i$ via representation-theoretic data and spectral sequence differentials.
• Applies Kronholm-type shift theorems to restrict possible shifts in the cohomology decomposition, particularly in the context of $C_2$-equivariant spaces.
• Leverages the non-Noetherian structure of $\mathbb{M}_2$, including elements $\rho \in H^{1,1}$, $\tau \in H^{0,1}$, and $\theta \in H^{0,-2}$, to describe the ring structure and its action on cohomology.

Experimental results

Research questions

• RQ1 What is the $RO(C_2)$-graded Bredon cohomology of the real Grassmannian $\operatorname{Gr}_k(\mathbb{R}^{n,1})$ as a free module over $\mathbb{M}_2$?
• RQ2 How do the weights $b_i$ in the decomposition $H^{\bullet,\bullet}(\operatorname{Gr}_k(\mathbb{R}^{n,1})) = \bigoplus_i \Sigma^{a_i,b_i}\mathbb{M}_2$ depend on the partition indexing the Schubert cells?
• RQ3 What are the nontrivial differentials in the equivariant cellular spectral sequence for $\operatorname{Gr}_2(\mathbb{R}^{n,2})$, and how do they affect the final cohomology structure?
• RQ4 How does the perp map $P$ induce an isomorphism on cohomology, and how can this duality be used to constrain the image of generators under the forgetful map?
• RQ5 What is the $RO(C_2)$-graded cohomology of the infinite Grassmannian $\operatorname{Gr}_k(\mathbb{R}^{\infty,1})$ and its complex analogs?

Key findings

• The $RO(C_2)$-graded Bredon cohomology of $\operatorname{Gr}_k(\mathbb{R}^{n,1})$ is computed as a free $\mathbb{M}_2$-module with explicit weights $b_i$ determined by the Schubert cell decomposition and representation-theoretic data.
• For $\operatorname{Gr}_2(\mathbb{R}^{4,1})$, the cohomology is $H^{\bullet,\bullet}(\operatorname{Gr}_2(\mathbb{R}^{4,1})) = \mathbb{M}_2 \oplus \Sigma^{1,1}\mathbb{M}_2 \oplus \Sigma^{2,1}\mathbb{M}_2$, showing a nontrivial weight distribution despite the non-equivariant singular cohomology being concentrated in degree 4.
• The perp map induces an isomorphism on cohomology, sending $[\Omega_\lambda]$ to $[\Omega_{\lambda^T}]$, which allows duality arguments to rule out certain generator images under the forgetful map.
• The spectral sequence for the Schubert cell filtration has nontrivial differentials, and these are managed via a theorem restricting Kronholm shifts, enabling explicit computation of the final cohomology.
• The cohomology of the infinite Grassmannian $\operatorname{Gr}_k(\mathbb{R}^{\infty,1})$ is computed as a direct limit of finite cases, yielding a free $\mathbb{M}_2$-module with weights determined by the same pattern as in the finite case.
• For complex Grassmannians, the cohomology structure is analogous, with $H^{\bullet,\bullet}(\operatorname{Gr}_k(\mathbb{C}^{n,q}))$ being a free $\mathbb{M}_2$-module, and the perp map duality holds in the complex setting as well.