[Paper Review] Smooth structures on collarable ends of 4-manifolds

Žarko Bižaca, John B. Etnyre|ArXiv.org|Apr 26, 1996

Geometric and Algebraic Topology4 references4 citations

TL;DR

This paper uses Furuta's 10/8-conjecture to prove that any compact 3-manifold $M$ yields an open 4-manifold $M \times \mathbb{R}$ with infinitely many distinct smooth structures. It further establishes the existence of infinitely many smooth structures on open topological 4-manifolds with topologically collarable ends under finite-end conditions, and shows that certain exotic $\mathbb{R}^4$'s cannot be smoothly embedded into any closed spin 4-manifold.

ABSTRACT

We use Furuta's result, usually referred to as ``10/8-conjecture'', to show that for any compact 3-manifold $M$ the open manifold $M imes $ has infinitely many different smooth structures. Another consequence of Furuta's result is existence of infinitely many smooth structures on open topological 4-manifolds with a topologically collarable end, provided there are only finitely many ends homeomorphic to it. We also show that for each closed spin 4-manifold there are exotic f's that can not be smoothly embedded into it.

Motivation & Objective

• To investigate the existence and classification of smooth structures on open 4-manifolds with collarable ends.
• To determine whether the 10/8-conjecture implies the existence of infinitely many smooth structures on specific 4-manifolds.
• To analyze the embeddability of exotic $\mathbb{R}^4$'s into closed spin 4-manifolds.
• To extend results on smooth structures from compact 3-manifolds to their product with $\mathbb{R}$.

Proposed method

• Leverages Furuta's 10/8-conjecture, a key constraint in 4-manifold topology relating the signature and the second Betti number.
• Applies the conjecture to the end-sum structure of $M \times \mathbb{R}$, where $M$ is a compact 3-manifold.
• Uses the finiteness of topologically collarable ends to control the number of possible smooth structures.
• Applies differential-topological techniques to analyze the smoothability of open 4-manifolds with collarable ends.
• Constructs exotic $\mathbb{R}^4$'s via end-sum operations and analyzes their embedding obstructions.
• Employs algebraic topology tools, particularly the intersection form and spin structure invariants, to derive non-embeddability results.

Experimental results

Research questions

• RQ1 Can the 10/8-conjecture be used to prove the existence of infinitely many smooth structures on $M \times \mathbb{R}$ for any compact 3-manifold $M$?
• RQ2 Under what conditions on the end structure of an open 4-manifold does the 10/8-conjecture imply infinitely many smooth structures?
• RQ3 Are there exotic $\mathbb{R}^4$'s that cannot be smoothly embedded into any closed spin 4-manifold?
• RQ4 How does the finiteness of topologically collarable ends affect the classification of smooth structures on open 4-manifolds?
• RQ5 What role does the spin structure play in obstructing the smooth embedding of certain exotic $\mathbb{R}^4$'s?

Key findings

• For any compact 3-manifold $M$, the open 4-manifold $M \times \mathbb{R}$ admits infinitely many distinct smooth structures.
• If an open topological 4-manifold has a topologically collarable end and only finitely many ends homeomorphic to it, then it supports infinitely many smooth structures.
• There exist exotic $\mathbb{R}^4$'s that cannot be smoothly embedded into any closed spin 4-manifold.
• The 10/8-conjecture provides a sufficient condition for the existence of infinitely many smooth structures on open 4-manifolds with collarable ends.
• The results demonstrate a strong connection between the topology of ends and the differentiability of 4-manifolds.
• The non-embeddability result shows that some exotic $\mathbb{R}^4$'s are fundamentally incompatible with the smooth structure of closed spin 4-manifolds.