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A generalization of moment-angle manifolds with non-contractible orbit spaces

Li Yu

Abstract

We generalize the notion of moment-angle manifold over a simple convex polytope to an arbitrary nice manifold with corners. For a nice manifold with corners Q, we first compute the stable decomposition of the moment-angle manifold Z_Q via a construction called rim-cubicalization of Q. From this, we derive a formula to compute the integral cohomology group of Z_Q via the strata of Q. This generalizes the Hochster's formula for the moment-angle manifold over a simple convex polytope. Moreover, we obtain a description of the integral cohomology ring of Z_Q using the idea of partial diagonal maps. In addition, we define the notion of polyhedral product of a sequence of based CW-complexes over Q and obtain similar results for these spaces as we do for Z_Q. Using this general construction, we can compute the equivariant cohomology ring of Z_Q with respect to its canonical torus action from the Davis-Januszkiewicz space of Q. The result leads to the definition of a new notion called the topological face ring of Q, which generalizes the notion of face ring of a simple polytope. Meanwhile, we obtain some parallel results for the real moment-angle manifold RZ_Q over Q.

A generalization of moment-angle manifolds with non-contractible orbit spaces

Abstract

We generalize the notion of moment-angle manifold over a simple convex polytope to an arbitrary nice manifold with corners. For a nice manifold with corners Q, we first compute the stable decomposition of the moment-angle manifold Z_Q via a construction called rim-cubicalization of Q. From this, we derive a formula to compute the integral cohomology group of Z_Q via the strata of Q. This generalizes the Hochster's formula for the moment-angle manifold over a simple convex polytope. Moreover, we obtain a description of the integral cohomology ring of Z_Q using the idea of partial diagonal maps. In addition, we define the notion of polyhedral product of a sequence of based CW-complexes over Q and obtain similar results for these spaces as we do for Z_Q. Using this general construction, we can compute the equivariant cohomology ring of Z_Q with respect to its canonical torus action from the Davis-Januszkiewicz space of Q. The result leads to the definition of a new notion called the topological face ring of Q, which generalizes the notion of face ring of a simple polytope. Meanwhile, we obtain some parallel results for the real moment-angle manifold RZ_Q over Q.

Paper Structure

This paper contains 12 sections, 33 theorems, 200 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Stable Decomposition of $\mathcal{Z}_Q$
  3. Rim-cubicalization of $Q$ in $Q\times [0,1]^m$
  4. Embedding $\mathcal{Z}_Q$ into $Q\times (D^2)^m$
  5. Viewing $\mathcal{Z}_Q$ as a colimit of CW-complexes
  6. Stable decomposition of $\mathcal{Z}_Q$
  7. Cohomology ring structure of $\mathcal{Z}_Q$
  8. Polyhedral product over a nice manifold with corners
  9. The case of $(\mathbb{X},\mathbb{A})^Q$ with each $X_j$ contractible
  10. The case of $(\mathbb{X},\mathbb{A})^Q$ with each $A_j$ contractible
  11. Equivariant cohomology ring of $\mathcal{Z}_Q$ and $\mathbb R\mathcal{Z}_Q$
  12. Generalizations

Key Result

Theorem 1.1

Let $Q$ be a nice manifold with corners with facets $F_1,\cdots, F_m$. There is a homotopy equivalence where $\bigvee$ denotes the wedge sum and $\mathbf{\Sigma}$ denotes the reduced suspension.

Figures (2)

  • Figure 1: Rim-cubicalization of $Q$ in $Q\times [0,1]^m$
  • Figure 2: Isotopy from $C^n_k(-1)$ to $C^n_k(0)$

Theorems & Definitions (71)

  • Theorem 1.1
  • Corollary 1.2
  • Remark 1.3
  • Theorem 1.4
  • Theorem 1.5: Corollary \ref{['Cor:Real-Moment']}
  • Definition 1.6: Topological Face Ring
  • Theorem 1.7
  • Remark 1.8
  • Lemma 2.1
  • proof
  • ...and 61 more