[Paper Review] Corks with large shadow-complexity and exotic 4-manifolds

Hironobu Naoe|arXiv (Cornell University)|Nov 14, 2017

Geometric and Algebraic Topology17 references3 citations

TL;DR

This paper constructs an infinite family of Mazur-type corks $C_{n,k}$ with special shadow-complexity bounded between $2n$ and $O(n^{3/2})$, demonstrating that exotic 4-manifolds can have arbitrarily large shadow-complexity. Using these corks, it further constructs exotic pairs $(W_{n,k}, W'_{n,k})$ with the same complexity bounds, resolving questions about the minimal complexity of exotic 4-manifolds with boundary.

ABSTRACT

We construct an infinite family $\{ C_{n,k}\}_{k=1}^{\infty}$ of corks of Mazur type satisfying $2n\leq \mathrm{sc}^{\mathrm{sp}}(C_{n,k})\leq O(n^{3/2})$ for any positive integer $n$. Furthermore, using these corks, we construct an infinite family $\{(W_{n,k},W'_{n,k})\}_{k=1}^{\infty}$ of exotic pairs of $4$-manifolds with boundary whose special shadow-complexities satisfy the above inequalities. We also discuss exotic pairs with small shadow-complexity.

Motivation & Objective

• To construct an infinite family of Mazur-type corks with controlled, large special shadow-complexity.
• To demonstrate that exotic 4-manifolds with boundary can have arbitrarily high shadow-complexity.
• To resolve the minimal special shadow-complexity of exotic pairs of 4-manifolds with boundary.
• To provide explicit upper and lower bounds on the special shadow-complexity of both corks and exotic pairs.

Proposed method

• The construction uses Kirby calculus and handlebody decompositions to define corks $C_{n,k}$ with $n+1$ 1-handles and $n+2$ 2-handles.
• A shadow-complexity bound is derived by analyzing the number of true vertices in a special shadow polyhedron constructed from the Kirby diagram.
• The upper bound $D(n)$ is computed via a formula involving ceiling functions of square roots related to $n$ and $\pi^2$.
• The lower bound $2n$ is established by analyzing the boundary 3-manifold's shadow-complexity using gleam conditions and Theorem 2.8.
• The exotic pairs $(W_{n,k}, W'_{n,k})$ are formed by attaching a 2-handle with framing $-1$ or $0$ to the cork $C_{n,k}$, yielding distinct smooth structures.
• The special shadow-complexity of the exotic pairs is bounded by the maximum of the individual complexities, derived from the shadow constructions of $W_{n,k}$ and $W'_{n,k}$.

Experimental results

Research questions

• RQ1 What is the minimal special shadow-complexity of exotic pairs of 4-manifolds with boundary?
• RQ2 Can there exist corks or exotic 4-manifolds with arbitrarily large shadow-complexity?
• RQ3 What is the relationship between the shadow-complexity of a cork and the complexity of the exotic 4-manifolds it generates?
• RQ4 How does the boundary 3-manifold's complexity relate to the special shadow-complexity of the 4-manifold?
• RQ5 Can the special shadow-complexity of exotic pairs be bounded below by a function of $n$?

Key findings

• An infinite family of Mazur-type corks $\{C_{n,k}\}_{k=1}^\infty$ is constructed with $2n \leq \mathrm{sc}^{\mathrm{sp}}(C_{n,k}) \leq D(n)$, where $D(n)$ is explicitly given by a piecewise formula involving ceiling functions.
• The upper bound $D(n)$ is asymptotically $O(n^{3/2})$, showing that shadow-complexity grows subquadratically with $n$.
• The exotic pairs $(W_{n,k}, W'_{n,k})$ have special shadow-complexity satisfying $2n \leq \mathrm{sc}^{\mathrm{sp}}(W_{n,k}, W'_{n,k}) \leq D(n)$, confirming large complexity in exotic structures.
• The minimal special shadow-complexity of exotic pairs of 4-manifolds with boundary is shown to be either 1 or 2, resolving a question posed by Costantino.
• The boundary 3-manifold of $W'_{n,k}$ has special shadow-complexity $2n$, which directly contributes to the lower bound on the 4-manifold's complexity.
• The construction confirms that there exist exotic 4-manifolds with boundary whose special shadow-complexity can be made arbitrarily large by increasing $n$.