[Paper Review] Perfect Morse functions and exotic S^2 x S^2's

Jacob Rasmussen|arXiv (Cornell University)|May 25, 2010

Geometric and Algebraic Topology14 references3 citations

TL;DR

This paper investigates the existence of perfect Morse functions on smooth manifolds homeomorphic to S²×S² with nonvanishing Ozsváth-Szabó invariants. Using gauge-theoretic invariants and Morse theory, it proves that such manifolds cannot admit perfect Morse functions, though the existence of the assumed manifolds remains uncertain, leading the author to withdraw the paper.

ABSTRACT

The main theorem of the paper shows that a smooth manifold which is homeomorphic to S^2xS^2 and has nonvanishing Ozsvath-Szabo invariant does not admit a perfect Morse function. I am withdrawing the paper because it is unclear to me if such a manifold exists.

Motivation & Objective

• To determine whether smooth manifolds homeomorphic to S²×S² with nonvanishing Ozsváth-Szabó invariants can admit perfect Morse functions.
• To explore the topological and smooth constraints imposed by gauge-theoretic invariants on Morse-theoretic structures.
• To assess the consistency of perfect Morse functions with the differential topology of exotic S²×S² structures.

Proposed method

• Application of Ozsváth-Szabó invariants as smooth invariants distinguishing exotic S²×S² manifolds.
• Use of Morse theory to analyze the critical point structure of smooth functions on 4-manifolds.
• Analysis of perfect Morse functions—those with the minimum number of critical points—on 4-manifolds homeomorphic to S²×S².
• Combining gauge-theoretic constraints with Morse-theoretic obstructions to derive non-existence results.
• Logical deduction that nonvanishing Ozsváth-Szabó invariants obstruct the existence of perfect Morse functions.

Experimental results

Research questions

• RQ1 Can a smooth manifold homeomorphic to S²×S² with nonvanishing Ozsváth-Szabó invariant admit a perfect Morse function?
• RQ2 What topological obstructions arise from the interplay between gauge invariants and Morse theory in 4-manifolds?
• RQ3 Do exotic S²×S² structures necessarily fail to support perfect Morse functions due to their smooth invariants?

Key findings

• A smooth manifold homeomorphic to S²×S² with nonvanishing Ozsváth-Szabó invariant cannot admit a perfect Morse function.
• The obstruction arises from the conflict between the minimal critical point structure of perfect Morse functions and the nontrivial gauge-theoretic invariants.
• The existence of such a manifold remains uncertain, casting doubt on the foundational assumption of the argument.
• The paper's conclusion is contingent on the existence of a manifold satisfying the invariance condition, which is currently unverified.
• Due to this foundational uncertainty, the author withdrew the paper, acknowledging the result's conditional nature.