Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On smooth approximation of integral cycles mod 2

Gianmarco Caldini

TL;DR

The paper extends smooth approximation results to unoriented settings by proving that any mod 2 integral m-cycle representing a class τ ∈ H_m(𝓜,ℤ2) can be approximated in flat norm by a smooth m-submanifold Σ outside an (m−n−1)-skeleton, with mass nearly that of the original cycle and with a controlled singular set whose codimension is sharp in full generality. The proof combines mod 2 Thom–Pontryagin techniques and a refined homotopy analysis of the Thom space T(γ^n), exploiting a 2n–equivalence to a product of Eilenberg–MacLane spaces to realize the Poincaré dual of τ via a Thom map, followed by a geometric measure theory squeezing argument to produce a smooth approximation S′ with mass close to the target. If τ admits a smooth embedded representative, the approximant Σ can be taken smooth with no singularities. The results yield optimal codimension estimates and clarify the unoriented Plateau problem in the mod 2 setting, including sharpness arguments grounded in stable map theory and obstructions identified by Grant–Szűcs.

Abstract

We prove that every mod 2 integral cycle $T$ in a Riemannian manifold $\mathcal{M}$ can be approximated in flat norm by a cycle which is a smooth submanifold $Σ$ of nearly the same area, up to a singular set of codimension 3; in addition, this estimate on the singular set can be refined depending on the codimension of the cycle. Moreover, if the mod 2 homology class $τ$ admits a smooth embedded representative, then $Σ$ can be chosen free of singularities. This article provides the unoriented version of the smooth approximation theorem for integral cycles.

On smooth approximation of integral cycles mod 2

TL;DR

The paper extends smooth approximation results to unoriented settings by proving that any mod 2 integral m-cycle representing a class τ ∈ H_m(𝓜,ℤ2) can be approximated in flat norm by a smooth m-submanifold Σ outside an (m−n−1)-skeleton, with mass nearly that of the original cycle and with a controlled singular set whose codimension is sharp in full generality. The proof combines mod 2 Thom–Pontryagin techniques and a refined homotopy analysis of the Thom space T(γ^n), exploiting a 2n–equivalence to a product of Eilenberg–MacLane spaces to realize the Poincaré dual of τ via a Thom map, followed by a geometric measure theory squeezing argument to produce a smooth approximation S′ with mass close to the target. If τ admits a smooth embedded representative, the approximant Σ can be taken smooth with no singularities. The results yield optimal codimension estimates and clarify the unoriented Plateau problem in the mod 2 setting, including sharpness arguments grounded in stable map theory and obstructions identified by Grant–Szűcs.

Abstract

We prove that every mod 2 integral cycle in a Riemannian manifold can be approximated in flat norm by a cycle which is a smooth submanifold of nearly the same area, up to a singular set of codimension 3; in addition, this estimate on the singular set can be refined depending on the codimension of the cycle. Moreover, if the mod 2 homology class admits a smooth embedded representative, then can be chosen free of singularities. This article provides the unoriented version of the smooth approximation theorem for integral cycles.

Paper Structure

This paper contains 15 sections, 21 theorems, 55 equations.

Table of Contents

  1. Introduction
  2. Main theorem and corollaries
  3. Motivation and overview of the proof
  4. Notation and preliminary results
  5. Theory of integral currents mod 2
  6. Homotopy theory and cobordism
  7. Regular neighborhoods and maps
  8. Neighborhoods of skeleta
  9. Squeezing maps
  10. Geometric measure theory propositions
  11. Codimension 2 smooth approximation
  12. Squeezing lemma
  13. Proof of the theorem
  14. Optimality of the construction
  15. Homotopy type of $T(\gamma^n)$

Key Result

Theorem 1.1

Let $\mathcal{M}$ be a connected smooth closed (not necessarily orientable) Riemannian manifold of dimension $m+n$. Let $\varepsilon > 0$, $\tau$ an $m$-dimensional homology class in $H_{m}(\mathcal{M}, \mathbb{Z}_2)$, and $T$ an integral cycle mod 2 representing $\tau$. Then there is a smooth trian

Theorems & Definitions (37)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Definition 1.6: Smooth representability
  • Theorem 1.7: Absence of Lavrentiev gaps
  • Remark 1.8
  • Theorem 1.9: Approximation by cycles with prescribed singular sets
  • Remark 1.10
  • Corollary 2.1: Spanier
  • ...and 27 more