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Homotopy commutativity in quasitoric manifolds

Sho Hasui, Daisuke Kishimoto, Yichen Tong, Mitsunobu Tsutaya

TL;DR

This work characterizes when the loop space of a quasitoric manifold is homotopy commutative by tying the property to the base polytope being a product of $3$-simplices, $(\Delta^3)^n$, and a parity condition on the characteristic matrix. It provides a loop-space decomposition $\Omega M \simeq T^{m-n} \times \Omega Z_{K(P)}$ and reduces the problem to Whitehead/Samelson product computations, extending Barrat–James–Stein methods to this setting. The authors construct an infinite family $M(k,n)$ of generalized Bott manifolds over $(\Delta^3)^n$, prove cohomological rigidity (distinct $k$ yield non-homotopy-equivalent manifolds) and establish that $\Omega M(k,n)$ is homotopy commutative precisely when $k$ is even. These results extend classical results on loop space commutativity, provide atomic examples over $(\Delta^3)^n$, and illuminate how combinatorial data determine higher-homotopy structure in quasitoric geometry.

Abstract

We prove that the loop space of a quasitoric manifold is homotopy commutative if and only if the underlying polytope is a product of $3$-simplices $(Δ^3)^n$ and the characteristic matrix is equivalent to a matrix of certain type. Quasitoric manifolds over $(Δ^3)^n$ include generalized Bott manifolds, and we also construct an infinite family of homotopy nonequivalent generalized Bott manifolds over $(Δ^3)^n$, only half of them have homotopy commutative loop spaces. In particular, for each $n\ge 2$, there are infinitely many homotopy types in $6n$-dimensional quasitoric manifolds having homotopy (non)commutative loop spaces.

Homotopy commutativity in quasitoric manifolds

TL;DR

This work characterizes when the loop space of a quasitoric manifold is homotopy commutative by tying the property to the base polytope being a product of -simplices, , and a parity condition on the characteristic matrix. It provides a loop-space decomposition and reduces the problem to Whitehead/Samelson product computations, extending Barrat–James–Stein methods to this setting. The authors construct an infinite family of generalized Bott manifolds over , prove cohomological rigidity (distinct yield non-homotopy-equivalent manifolds) and establish that is homotopy commutative precisely when is even. These results extend classical results on loop space commutativity, provide atomic examples over , and illuminate how combinatorial data determine higher-homotopy structure in quasitoric geometry.

Abstract

We prove that the loop space of a quasitoric manifold is homotopy commutative if and only if the underlying polytope is a product of -simplices and the characteristic matrix is equivalent to a matrix of certain type. Quasitoric manifolds over include generalized Bott manifolds, and we also construct an infinite family of homotopy nonequivalent generalized Bott manifolds over , only half of them have homotopy commutative loop spaces. In particular, for each , there are infinitely many homotopy types in -dimensional quasitoric manifolds having homotopy (non)commutative loop spaces.
Paper Structure (4 sections, 25 theorems, 71 equations)

This paper contains 4 sections, 25 theorems, 71 equations.

Table of Contents

  1. Introduction
  2. Loop space decomposition
  3. Computation of Whitehead products
  4. Quasitoric manifolds over $(\Delta^3)^n$

Key Result

Theorem 1.1

The loop space of a quasitoric manifold over a simple polytope $P$ is homotopy commutative if and only if $P=(\Delta^3)^n$ and the characteristic matrix is equivalent to for $a_{ij}\in\mathbb{Z}^3$ such that where the facets of $(\Delta^3)^n$ are ordered as in Section Quasitoric manifold.

Theorems & Definitions (46)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 2.1
  • proof
  • Proposition 2.2
  • Proposition 2.3
  • Proposition 2.4
  • proof
  • Lemma 2.5
  • proof
  • ...and 36 more