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On the integral cohomology of real toric manifolds

Feifei Fan

TL;DR

The paper develops integral cohomology tools for real toric spaces M(K,Λ) by constructing explicit DGA models for the real moment-angle complex and its quotients, linking cohomology to combinatorial data of K and Λ. It provides an isomorphism of cohomology groups with a combinatorial DGA, extends additive structure results to PL-sphere cases, and introduces a ring structure modulo 2 via a novel $*$-product on a combinatorial group G*(K,Λ). The work also establishes a spin$^c$ criterion and builds a framework for computing equivariant and ordinary cohomology through shellability and subdivision arguments, ultimately yielding a practical, combinatorial description of integral cohomology for real toric manifolds and small covers. The results unify and extend Cai–Choi and related approaches, offering computational tools and structural insights with potential applications in toric topology and related fields.

Abstract

Real toric manifolds are the real loci of nonsingular complete toric varieties. In this paper, we calculate the integral cohomology groups of real toric manifolds in terms of the combinatorial data contained in the underlying simplicial fans, generalizing a formula due to Cai and Choi. We also present a combinatorial formula to describe the ring structure, modulo the ideal consisting of elements of order $2$. As an application, a combinatorial-algebraic criterion is established for when a real toric manifold is spin$^c$.

On the integral cohomology of real toric manifolds

TL;DR

The paper develops integral cohomology tools for real toric spaces M(K,Λ) by constructing explicit DGA models for the real moment-angle complex and its quotients, linking cohomology to combinatorial data of K and Λ. It provides an isomorphism of cohomology groups with a combinatorial DGA, extends additive structure results to PL-sphere cases, and introduces a ring structure modulo 2 via a novel -product on a combinatorial group G*(K,Λ). The work also establishes a spin criterion and builds a framework for computing equivariant and ordinary cohomology through shellability and subdivision arguments, ultimately yielding a practical, combinatorial description of integral cohomology for real toric manifolds and small covers. The results unify and extend Cai–Choi and related approaches, offering computational tools and structural insights with potential applications in toric topology and related fields.

Abstract

Real toric manifolds are the real loci of nonsingular complete toric varieties. In this paper, we calculate the integral cohomology groups of real toric manifolds in terms of the combinatorial data contained in the underlying simplicial fans, generalizing a formula due to Cai and Choi. We also present a combinatorial formula to describe the ring structure, modulo the ideal consisting of elements of order . As an application, a combinatorial-algebraic criterion is established for when a real toric manifold is spin.

Paper Structure

This paper contains 8 sections, 35 theorems, 182 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. A DGA model for $\mathbb{R}\mathcal{Z}_K$
  4. A DGA model for $M(K,\Lambda)$
  5. The equivariant cohomology of $\mathbb{R}\mathcal{Z}_K$
  6. Real toric spaces associated to shellable complexes
  7. Polytopal spheres and subdivisions of PL spheres
  8. The $*$-product on $\mathcal{G}^*(K,\Lambda)$

Key Result

Theorem 2.1

Let $\Lambda:(\mathbb{Z}/2)^m\to (\mathbb{Z}/2)^n$ be a characteristic function over $K$. Then $\mathcal{R}(K,\Lambda)\subset\mathcal{R}_K$ is the submodule and there is a graded $\Bbbk$-linear isomorphism for any coefficient ring $\Bbbk$

Theorems & Definitions (61)

  • Theorem 2.1
  • Theorem 2.2
  • Remark 2.3
  • Theorem 2.4: DJ91
  • Theorem 2.5
  • Theorem 2.6
  • Corollary 2.7
  • proof
  • Proposition 2.8
  • Proposition 2.9
  • ...and 51 more