Noriyuki Hamada - Exotic 4-manifolds with signature zero

Abstract

We will talk about our recent construction of innitely many simply- connected closed exotic symplectic 4-manifolds with signature zero, which are homeomorphic but not diffeomorphic to \#2k+1(S2S2) and $\#2l+1(CP2\#CP2)$ for each k  5 and l  4, respectively. Notably, they provide the smallest such exotic 4-manifolds known to date. Moreover, they populate some new lattices in symplectic geography of simply-connected minimal symplectic 4-manifolds of general type. We build our examples from scratch: rst, construct fairly special small Lefschetz brations over tori with signature zero, take ber sum with the trivial bration over a torus, and then apply surgeries along Lagrangian tori to kill 1. This is joint work with Inanc Baykur (University of Massachusetts Amherst).

This is a copy of Konferenz Hamada - Exotic 4-manifolds with signature zero via Lefschetz fibrations

Introduction

We assume our manifolds to be connect4d, cosed oriented. Today we are interested in 4-mfd with signature 0 (i.e. they are connceted sums of tori or a product of complex projected planes) Question: Are there exotic maniolds of sigature 0?

Main result

Boykur-H. ‘23: For specific conditions we can find inf. many exatic structures that are exotic to a sign 0 mfd.

Definitions

A signature is calculated by taking two surfaces from the second homology of the $4$-space, counting their intersections, writing them down in a matrix and taking the signature.

Freedmans theorem uses the signature as well as the euler characteristic and parity to classify $4$-manifolds up to homeomorphism.

Generally the $4$-mfd can be classified by “simple prime-like mfd” and their connected sums. If besides Freedman, Roklin, Donaldson we use “the 11/8 conjecture” then the $4$-dimensional mfd are nicely classified.

For mfd. with signature 0 we dont need the conjecture to make a good classification.

Examples for exotic structures

Many authors tried to discover $4$-mfd.

Symplectic Geography

Assume $X$ is a symplectic $4$-mfd. This is a property from algebrai geometry.

Geography problem: For given $x,y$ determine whether there exists a $4$-mfd with two invariants being given by $x,y$.

Construction of the main theorem

1. We find a model manifold with some conditions
2. We apply surgeries to create new mfds $X_n$
3. If they have ${\pi }_1(X_n)=1$ then we are done.

In the publication the model manifold is generated using lefschetz fibrations instead of algebraic surfaces.