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Smoothly knotted and topologically unknotted nullhomologous surfaces in 4-manifolds

Rafael Torres

Abstract

We point out that recent constructions of inequivalent smooth structures yield a manufacturing procedure of infinite sets of pairwise smoothly non-isotopic nullhomologous 2-tori and spheres inside a myriad of 4-manifolds. The corresponding infinite set consists of topologically isotopic surfaces that topologically bound a handlebody in several instances.

Smoothly knotted and topologically unknotted nullhomologous surfaces in 4-manifolds

Abstract

We point out that recent constructions of inequivalent smooth structures yield a manufacturing procedure of infinite sets of pairwise smoothly non-isotopic nullhomologous 2-tori and spheres inside a myriad of 4-manifolds. The corresponding infinite set consists of topologically isotopic surfaces that topologically bound a handlebody in several instances.

Paper Structure

This paper contains 12 sections, 11 theorems, 68 equations.

Table of Contents

  1. Introduction and main result
  2. Acknowledgements:
  3. Tools and technology
  4. Torus surgeries
  5. Surgeries along loops and 2-spheres
  6. Three pairs of geometrically dual 2-tori in the 4-torus, some useful diffeomorphisms and a homeomorphism
  7. Examples out of Reverse-Engineering of 4-manifolds
  8. Fintushel-Stern's invariant of 2-tori
  9. Proof of main results.
  10. Proof of Theorem \ref{['Theorem A']}
  11. Proof of Theorem \ref{['Theorem FHMT']}
  12. Proof of Theorem \ref{['Theorem D']}

Key Result

Theorem 1

Let $X$ be a closed symplectic 4-manifold that contains a 2-torus $T$, which is either symplectic or homologically essential and Lagrangian. Suppose that the normal bundle of $T$ is trivial, and that both $X$ and $X\setminus \nu(T)$ are simply connected. $\bullet$ There is an infinite set of topolog embedded in $X\#S^2\times S^2$ that satisfy $[S_i] = 0 \in H_2(X\#S^2\times S^2; \mathbb{Z})$ and 2

Theorems & Definitions (20)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Definition 1
  • Definition 2
  • Definition 3
  • Lemma 1
  • Theorem \oldthetheorem
  • proof
  • ...and 10 more