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Geometric obstructions for $ξ$-fillings of 3-manifolds

Daniel Galvin, Peter Teichner, Simona Veselá

TL;DR

This work develops a geometric obstruction theory for realizing a given normal 1-type $\xi$ on a closed 3-manifold as the boundary of a $\xi$-4-manifold. It introduces primary, secondary, and tertiary obstructions realized through geometric invariants derived from Wall's self-intersection concepts, refined to land in $H_1(\pi;\mathbb{Z}/2)$, and ties these to James spectral sequence differentials. The authors extend Kreck's modified surgery framework to describe when $\xi$-fillings exist and show that the vanishing of the geometric tertiary obstruction $\mathfrak{ter}^{geo}$ is equivalent to the existence of a $\xi$-filling, with a precise comparison to the algebraic tertiary invariant. Key ingredients include a non-orientable extension of the James spectral sequence, refined intersection invariants $\mu_J$ and $\lambda_J$, and secondary Wu-type formulas, culminating in a complete obstruction theory for null-bordism in this setting. The results connect 3-manifold fillability to 4-manifold bordism through explicit geometric constructions and offer a framework for understanding higher-stage obstructions in the $\xi$-bordism context.

Abstract

We consider the realisation problem for normal 1-types of 4-manifolds with a given boundary. More precisely, given a normal 1-type $ξ$ and closed 3-dimensional $ξ$-manifold $Y$, does there exist a compact 4-dimensional $ξ$-manifold with boundary $Y$? We describe a three stage obstruction theory for the existence of such a 4-manifold, with our main contribution being a `tertiary' obstruction that we describe geometrically via Wall's quadratic self-intersection form.

Geometric obstructions for $ξ$-fillings of 3-manifolds

TL;DR

This work develops a geometric obstruction theory for realizing a given normal 1-type on a closed 3-manifold as the boundary of a -4-manifold. It introduces primary, secondary, and tertiary obstructions realized through geometric invariants derived from Wall's self-intersection concepts, refined to land in , and ties these to James spectral sequence differentials. The authors extend Kreck's modified surgery framework to describe when -fillings exist and show that the vanishing of the geometric tertiary obstruction is equivalent to the existence of a -filling, with a precise comparison to the algebraic tertiary invariant. Key ingredients include a non-orientable extension of the James spectral sequence, refined intersection invariants and , and secondary Wu-type formulas, culminating in a complete obstruction theory for null-bordism in this setting. The results connect 3-manifold fillability to 4-manifold bordism through explicit geometric constructions and offer a framework for understanding higher-stage obstructions in the -bordism context.

Abstract

We consider the realisation problem for normal 1-types of 4-manifolds with a given boundary. More precisely, given a normal 1-type and closed 3-dimensional -manifold , does there exist a compact 4-dimensional -manifold with boundary ? We describe a three stage obstruction theory for the existence of such a 4-manifold, with our main contribution being a `tertiary' obstruction that we describe geometrically via Wall's quadratic self-intersection form.
Paper Structure (27 sections, 58 theorems, 141 equations, 3 figures)

This paper contains 27 sections, 58 theorems, 141 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Conventions and notation
  4. Organisation
  5. Acknowledgements
  6. Modified Surgery
  7. Normal 1--types
  8. Obstruction theory for xi-structures
  9. Intersections and self intersections
  10. Preliminaries
  11. The form mudashj
  12. The forms mudashj and lambdadashj in relation to normal 1-types
  13. Secondary intersections
  14. Obstruction theory for filling 3-manifolds
  15. The primary obstruction
  16. ...and 12 more sections

Key Result

Theorem A

Let $\xi(\pi,w_1^{\pi},w_2^{\pi})\colon B\to BO$ be a normal 1-type with $\pi_2(B)=0$ and let $(Y,u_Y)$ be a 3-dimensional $\xi$-manifold. For the above filtration of $\Omega_3^{\xi}$ the corresponding geometric conditions are as follows: where ${{\mkern0.75mu\raisebox{-1.52ex}{$\mathchar '26$}\mkern -5.8mu\mu}}_{\raisebox{6pt}{$J$}}$ denotes a certain modification of Wall's self-intersection num

Figures (3)

  • Figure 1: A Kirby diagram for the manifold $X=((\mathbb{RP}^2\times S^2)\#(\mathbb{RP}^2\times S^2))_*$. The notation for the non-orientable 1-handles is due to Akbulut akbulut
  • Figure 2: The annulus $A_t$ described using a regular homotopy of $\gamma$, shown in sporadically dashed green, whereas the relevant parts of the Kirby diagram are shown in solid black. The guide for the single intersection point is shown in dashed purple. The handle slides are guided by the blue arrows. Only relevant portions of the Kirby diagram from \ref{['fig:rp2_x_s2_rp2_x_s2_kirby diagram']} are shown.
  • Figure 3: On the left: a Kirby diagram for the manifold $V_g$ if $g$ is orientable. On the right: a Kirby diagram for the manifold $V_g$ if $g$ is non-orientable. The notation for the 1-handles is due to Akbulut akbulut

Theorems & Definitions (143)

  • Theorem A
  • Theorem B
  • Remark 1.1
  • Theorem C
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • ...and 133 more