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<p>Let <i>M</i> be a closed, 3-connected, 8-dimensional smooth manifold. In this paper, we compute the concordance inertia group of the product manifold <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12044_2025_829_Article_IEq1.gif" Format="GIF" Height="18" Rendition="HTML" Resolution="72" Type="Linedraw" Width="59" /> </InlineMediaObject> <EquationSource Format="TEX">\(M\times \mathbb {S}^k\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>M</mi> <mo>×</mo> <msup> <mrow> <mi mathvariant="double-struck">S</mi> </mrow> <mi>k</mi> </msup> </mrow> </math></EquationSource> </InlineEquation> for <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12044_2025_829_Article_IEq2.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="81" /> </InlineMediaObject> <EquationSource Format="TEX">\(1\le k\le 14\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>1</mn> <mo>≤</mo> <mi>k</mi> <mo>≤</mo> <mn>14</mn> </mrow> </math></EquationSource> </InlineEquation> and classify all smooth manifolds homeomorphic to <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12044_2025_829_Article_IEq1.gif" Format="GIF" Height="18" Rendition="HTML" Resolution="72" Type="Linedraw" Width="59" /> </InlineMediaObject> <EquationSource Format="TEX">\(M\times \mathbb {S}^k\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>M</mi> <mo>×</mo> <msup> <mrow> <mi mathvariant="double-struck">S</mi> </mrow> <mi>k</mi> </msup> </mrow> </math></EquationSource> </InlineEquation> up to concordance for <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12044_2025_829_Article_IEq4.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="81" /> </InlineMediaObject> <EquationSource Format="TEX">\(1\le k\le 10\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>1</mn> <mo>≤</mo> <mi>k</mi> <mo>≤</mo> <mn>10</mn> </mrow> </math></EquationSource> </InlineEquation>. Moreover, we provide a diffeomorphism classification of smooth manifolds homeomorphic to <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12044_2025_829_Article_IEq5.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="62" /> </InlineMediaObject> <EquationSource Format="TEX">\(M\times \mathbb {S}^1,\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>M</mi> <mo>×</mo> <msup> <mrow> <mi mathvariant="double-struck">S</mi> </mrow> <mn>1</mn> </msup> <mo>,</mo> </mrow> </math></EquationSource> </InlineEquation> where <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12044_2025_829_Article_IEq6.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="109" /> </InlineMediaObject> <EquationSource Format="TEX">\(H^4(M;\mathbb {Z})=\mathbb {Z}.\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>H</mi> <mn>4</mn> </msup> <mrow> <mo stretchy="false">(</mo> <mi>M</mi> <mo>;</mo> <mi mathvariant="double-struck">Z</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mi mathvariant="double-struck">Z</mi> <mo>.</mo> </mrow> </math></EquationSource> </InlineEquation></p>

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Smooth structures on the product of 3-connected 8-manifolds with spheres

  • 期刊论文

  • 发表日期：

    2025年8月1日

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  • Ankur Sarkar

摘要

Let M be a closed, 3-connected, 8-dimensional smooth manifold. In this paper, we compute the concordance inertia group of the product manifold \(M\times \mathbb {S}^k\) M × S k for \(1\le k\le 14\) 1 ≤ k ≤ 14 and classify all smooth manifolds homeomorphic to \(M\times \mathbb {S}^k\) M × S k up to concordance for \(1\le k\le 10\) 1 ≤ k ≤ 10 . Moreover, we provide a diffeomorphism classification of smooth manifolds homeomorphic to \(M\times \mathbb {S}^1,\) M × S 1 , where \(H^4(M;\mathbb {Z})=\mathbb {Z}.\) H 4 ( M ; Z ) = Z .