Let <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$T(\gamma )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>T</mi> <mo stretchy="false">(</mo> <mi>γ</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> be the total space of the canonical line bundle <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\gamma $</EquationSource> <EquationSource Format="MATHML"><math> <mi>γ</mi> </math></EquationSource> </InlineEquation> over <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\mathbb {C}\textrm{P}^1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="double-struck">C</mi> <msup> <mtext>P</mtext> <mn>1</mn> </msup> </mrow> </math></EquationSource> </InlineEquation> and r an integer, which is divisible by an odd prime. We prove that <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$L_r^3\times T(\gamma )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mi>L</mi> <mi>r</mi> <mn>3</mn> </msubsup> <mo>×</mo> <mi>T</mi> <mrow> <mo stretchy="false">(</mo> <mi>γ</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> admits an infinite sequence of metrics of nonnegative sectional curvature with pairwise non-homeomorphic souls, where <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$L_r^3$</EquationSource> <EquationSource Format="MATHML"><math> <msubsup> <mi>L</mi> <mi>r</mi> <mn>3</mn> </msubsup> </math></EquationSource> </InlineEquation> is a 3-dimensional lens space with fundamental group of order r. Furthermore, we classify a class of non-simply connected 5-manifolds up to diffeomorphism and use this result to give first examples of manifolds N, which admit two complete metrics of nonnegative sectional curvature with souls S and <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$S'$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>S</mi> <mo>′</mo> </msup> </math></EquationSource> </InlineEquation> of codimension two such that S and <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$S'$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>S</mi> <mo>′</mo> </msup> </math></EquationSource> </InlineEquation> are diffeomorphic whereas the pairs (N, S) and <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$(N,S')$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>N</mi> <mo>,</mo> <msup> <mi>S</mi> <mo>′</mo> </msup> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> are not diffeomorphic. These results give solutions to two problems posed by Igor Belegradek, Slawomir Kwasik and Reinhard Schultz.

A classification of 5-dimensional manifolds, homogeneous souls of codimension two and non-diffeomorphic pairs

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2026年2月18日

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Sadeeb Simon Ottenburger

摘要

Let $T(\gamma )$ $T (γ)$ be the total space of the canonical line bundle $\gamma $ $γ$ over $\mathbb {C}\textrm{P}^1$ $C P^{1}$ and r an integer, which is divisible by an odd prime. We prove that $L_r^3\times T(\gamma )$ $L_{r}^{3} \times T (γ)$ admits an infinite sequence of metrics of nonnegative sectional curvature with pairwise non-homeomorphic souls, where $L_r^3$ $L_{r}^{3}$ is a 3-dimensional lens space with fundamental group of order r. Furthermore, we classify a class of non-simply connected 5-manifolds up to diffeomorphism and use this result to give first examples of manifolds N, which admit two complete metrics of nonnegative sectional curvature with souls S and $S'$ $S^{'}$ of codimension two such that S and $S'$ $S^{'}$ are diffeomorphic whereas the pairs (N, S) and $(N,S')$ $(N, S^{'})$ are not diffeomorphic. These results give solutions to two problems posed by Igor Belegradek, Slawomir Kwasik and Reinhard Schultz.