Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Geometric obstructions to fully ellipticity for families of manifolds with corners

Florian Thiry

TL;DR

The paper analyzes obstructions to turning elliptic operators on families of manifolds with corners into fully elliptic (Fredholm) operators. It develops a $C^*$-algebraic diagonal obstruction via Ind_{\partial} and couples this with a groupoid-based relative ellipticity framework using the tangent groupoid. The main results give explicit obstruction spaces in codimension 1 and a new codimension-2 combinatorial condition expressed through conormal homology with coefficients, linking $K$-theory to face-wise indices. These tools provide a structured method to determine when boundary perturbations suffice to achieve full ellipticity and point toward rational (torsion-free) reductions in higher codimensions. The construction relies on Monthubert groupoids, conormal-homology pairings, and six-term exact sequences to connect analytic indices with geometric-face data.

Abstract

In the present paper we study index theory for families of manifold with corners. In particular the K-theoretical obstruction for an elliptic operator to have a family (Fredholm) index. For codimension 1 corners (families with boundary) it gives an expected condition on indices associated to codimension 1 faces. The main theorem of this paper concern the codimension 2 case: in addition to the expected condition on indices associated to codimension 2 faces, there is an extra combinatoric condition implying the topology of the family base. This new condition is expressed in terms of conormal homology cycles with K-theoretical coefficients.

Geometric obstructions to fully ellipticity for families of manifolds with corners

TL;DR

The paper analyzes obstructions to turning elliptic operators on families of manifolds with corners into fully elliptic (Fredholm) operators. It develops a -algebraic diagonal obstruction via Ind_{\partial} and couples this with a groupoid-based relative ellipticity framework using the tangent groupoid. The main results give explicit obstruction spaces in codimension 1 and a new codimension-2 combinatorial condition expressed through conormal homology with coefficients, linking -theory to face-wise indices. These tools provide a structured method to determine when boundary perturbations suffice to achieve full ellipticity and point toward rational (torsion-free) reductions in higher codimensions. The construction relies on Monthubert groupoids, conormal-homology pairings, and six-term exact sequences to connect analytic indices with geometric-face data.

Abstract

In the present paper we study index theory for families of manifold with corners. In particular the K-theoretical obstruction for an elliptic operator to have a family (Fredholm) index. For codimension 1 corners (families with boundary) it gives an expected condition on indices associated to codimension 1 faces. The main theorem of this paper concern the codimension 2 case: in addition to the expected condition on indices associated to codimension 2 faces, there is an extra combinatoric condition implying the topology of the family base. This new condition is expressed in terms of conormal homology cycles with K-theoretical coefficients.
Paper Structure (12 sections, 26 theorems, 36 equations)

This paper contains 12 sections, 26 theorems, 36 equations.

Table of Contents

  1. Obstruction and diagonal index
  2. $C^*$ algebraic obstruction theorem
  3. Relative ellipticity
  4. Family of manifolds with corners
  5. Monthubert groupoids for families
  6. Monthubert groupoid for families
  7. Connections computation
  8. Pairing $K_*-H_*^{pcn}$
  9. Obstruction computation for embedded families in low codimension
  10. Boundary index for codimension 1 families globally embedded
  11. Boundary index for codimension 2 families globally embedded
  12. Appendix

Key Result

Theorem 1

For $X \longrightarrow B$ a codimension 1 family of manifold with embedded corners, $[\sigma_T]_0 \in K^0(\Gamma^{tan}_{b, \mathfrak{f}}(X))$, the following are equivalent: $Ind_{\partial}([\sigma_T]_0) = 0$ ssi $\forall g \in \mathcal{F}_1(X), Ind_1^g([{\sigma_T}_{\vert g}]_0) = 0$ in $K^1(B)$.

Theorems & Definitions (55)

  • Theorem 1: Vanishing of the boundary index in codimension 1
  • Theorem 1: Obstruction space estimation in codimension 2
  • Theorem 1: Vanishing of the boundary index in codimension 2
  • Lemma 1.1
  • Theorem 1.2
  • proof
  • Definition 1
  • Theorem 1.3
  • Remark 1
  • Remark 2
  • ...and 45 more