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

Abstract Let $$T(\gamma )$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>T</mml:mi> <mml:mo>(</mml:mo> <mml:mi>γ</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> be the total space of the canonical line bundle $$\gamma $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>γ</mml:mi> </mml:math> over $$\mathbb {C}\textrm{P}^1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>C</mml:mi> <mml:msup> <mml:mtext>P</mml:mtext> <mml:mn>1</mml:mn> </mml:msup> </mml:mrow> </mml:math> and r an integer, which is divisible by an odd prime. We prove that $$L_r^3\times T(\gamma )$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msubsup> <mml:mi>L</mml:mi> <mml:mi>r</mml:mi> <mml:mn>3</mml:mn> </mml:msubsup> <mml:mo>×</mml:mo> <mml:mi>T</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>γ</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> admits an infinite sequence of metrics of nonnegative sectional curvature with pairwise non-homeomorphic souls, where $$L_r^3$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>L</mml:mi> <mml:mi>r</mml:mi> <mml:mn>3</mml:mn> </mml:msubsup> </mml:math> 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 $$S'$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>S</mml:mi> <mml:mo>′</mml:mo> </mml:msup> </mml:math> of codimension two such that S and $$S'$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>S</mml:mi> <mml:mo>′</mml:mo> </mml:msup> </mml:math> are diffeomorphic whereas the pairs ( N , S ) and $$(N,S')$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>N</mml:mi> <mml:mo>,</mml:mo> <mml:msup> <mml:mi>S</mml:mi> <mml:mo>′</mml:mo> </mml:msup> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> are not diffeomorphic. These results give solutions to two problems posed by Igor Belegradek, Slawomir Kwasik and Reinhard Schultz.