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<p>We provide an approach to study exotic phenomena in relatively small 4-manifolds that captures many different exotic behaviors under one umbrella. These phenomena include exotic smooth structures on 4-manifolds with <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(b_2=1\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>b</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>, examples of strong corks, and exotic codimension-1 embeddings into <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(C P^2 \# - C P^2\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>C</mi> <msup> <mi>P</mi> <mn>2</mn> </msup> <mo>#</mo> <mo>-</mo> <mi>C</mi> <msup> <mi>P</mi> <mn>2</mn> </msup> </mrow> </math></EquationSource> </InlineEquation> that survive external stabilization. We also give a new way to detect a homeomorphism of a 4-manifold that is not topologically isotopic to any diffeomorphism and give lower bounds of relative genera of certain knots. Our primary tools are constraints on diffeomorphisms of 4-manifolds obtained from families Seiberg–Witten theory.</p>

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From diffeomorphisms to exotic phenomena in small 4-manifolds

  • 期刊论文

  • 发表日期：

    2026年3月10日

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  • Hokuto Konno,
  • Abhishek Mallick,
  • Masaki Taniguchi

摘要

We provide an approach to study exotic phenomena in relatively small 4-manifolds that captures many different exotic behaviors under one umbrella. These phenomena include exotic smooth structures on 4-manifolds with \(b_2=1\) b 2 = 1 , examples of strong corks, and exotic codimension-1 embeddings into \(C P^2 \# - C P^2\) C P 2 # - C P 2 that survive external stabilization. We also give a new way to detect a homeomorphism of a 4-manifold that is not topologically isotopic to any diffeomorphism and give lower bounds of relative genera of certain knots. Our primary tools are constraints on diffeomorphisms of 4-manifolds obtained from families Seiberg–Witten theory.