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• W4200403493 abstract "Iwase [Topology 42 (2003), pp. 701–713] has constructed two 16-dimensional manifolds <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper M 2> <mml:semantics> <mml:msub> <mml:mi>M</mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:annotation encoding=application/x-tex>M_2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper M 3> <mml:semantics> <mml:msub> <mml:mi>M</mml:mi> <mml:mn>3</mml:mn> </mml:msub> <mml:annotation encoding=application/x-tex>M_3</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with LS-category 3 which are counter-examples to Ganea’s conjecture: <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=c a t left-parenthesis upper M times upper S Superscript n Baseline right-parenthesis equals c a t upper M plus 1> <mml:semantics> <mml:mrow> <mml:mi>cat</mml:mi> <mml:mo>⁡<!-- ⁡ --></mml:mo> <mml:mo stretchy=false>(</mml:mo> <mml:mi>M</mml:mi> <mml:mo>×<!-- × --></mml:mo> <mml:msup> <mml:mi>S</mml:mi> <mml:mi>n</mml:mi> </mml:msup> <mml:mo stretchy=false>)</mml:mo> <mml:mo>=</mml:mo> <mml:mi>cat</mml:mi> <mml:mo>⁡<!-- ⁡ --></mml:mo> <mml:mi>M</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding=application/x-tex>operatorname {cat} (Mtimes S^n)=operatorname {cat} M+1</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. We show that the manifold <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper M 3> <mml:semantics> <mml:msub> <mml:mi>M</mml:mi> <mml:mn>3</mml:mn> </mml:msub> <mml:annotation encoding=application/x-tex>M_3</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a counter-example to the logarithmic law for the LS-category of the square of a manifold: <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=c a t left-parenthesis upper M times upper M right-parenthesis equals 2 c a t upper M> <mml:semantics> <mml:mrow> <mml:mi>cat</mml:mi> <mml:mo>⁡<!-- ⁡ --></mml:mo> <mml:mo stretchy=false>(</mml:mo> <mml:mi>M</mml:mi> <mml:mo>×<!-- × --></mml:mo> <mml:mi>M</mml:mi> <mml:mo stretchy=false>)</mml:mo> <mml:mo>=</mml:mo> <mml:mn>2</mml:mn> <mml:mi>cat</mml:mi> <mml:mo>⁡<!-- ⁡ --></mml:mo> <mml:mi>M</mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>operatorname {cat}(Mtimes M)=2operatorname {cat} M</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Also we construct a map of degree one <disp-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=f colon 2 left-parenthesis upper M 3 times upper S squared times upper S Superscript 14 Baseline right-parenthesis number-sign minus left-parenthesis upper M 2 times upper S squared times upper S Superscript 14 Baseline right-parenthesis right-arrow upper M 2 times upper M 3> <mml:semantics> <mml:mrow> <mml:mi>f</mml:mi> <mml:mo>:</mml:mo> <mml:mn>2</mml:mn> <mml:mo stretchy=false>(</mml:mo> <mml:msub> <mml:mi>M</mml:mi> <mml:mn>3</mml:mn> </mml:msub> <mml:mo>×<!-- × --></mml:mo> <mml:msup> <mml:mi>S</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mo>×<!-- × --></mml:mo> <mml:msup> <mml:mi>S</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mn>14</mml:mn> </mml:mrow> </mml:msup> <mml:mo stretchy=false>)</mml:mo> <mml:mi mathvariant=normal>#<!-- # --></mml:mi> <mml:mo>−<!-- − --></mml:mo> <mml:mo stretchy=false>(</mml:mo> <mml:msub> <mml:mi>M</mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:mo>×<!-- × --></mml:mo> <mml:msup> <mml:mi>S</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mo>×<!-- × --></mml:mo> <mml:msup> <mml:mi>S</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mn>14</mml:mn> </mml:mrow> </mml:msup> <mml:mo stretchy=false>)</mml:mo> <mml:mo stretchy=false>→<!-- → --></mml:mo> <mml:msub> <mml:mi>M</mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:mo>×<!-- × --></mml:mo> <mml:msub> <mml:mi>M</mml:mi> <mml:mn>3</mml:mn> </mml:msub> </mml:mrow> <mml:annotation encoding=application/x-tex>begin{equation*} f:2(M_3times S^2times S^{14})#-(M_2times S^2times S^{14})to M_2times M_3 end{equation*}</mml:annotation> </mml:semantics> </mml:math> </disp-formula> which reduces Rudyak’s conjecture to the question whether <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=c a t left-parenthesis upper M 2 times upper M 3 right-parenthesis greater-than-or-equal-to 5> <mml:semantics> <mml:mrow> <mml:mi>cat</mml:mi> <mml:mo>⁡<!-- ⁡ --></mml:mo> <mml:mo stretchy=false>(</mml:mo> <mml:msub> <mml:mi>M</mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:mo>×<!-- × --></mml:mo> <mml:msub> <mml:mi>M</mml:mi> <mml:mn>3</mml:mn> </mml:msub> <mml:mo stretchy=false>)</mml:mo> <mml:mo>≥<!-- ≥ --></mml:mo> <mml:mn>5</mml:mn> </mml:mrow> <mml:annotation encoding=application/x-tex>operatorname {cat}(M_2times M_3)ge 5</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and show that <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=c a t left-parenthesis upper M 2 times upper M 3 right-parenthesis greater-than-or-equal-to 4> <mml:semantics> <mml:mrow> <mml:mi>cat</mml:mi> <mml:mo>⁡<!-- ⁡ --></mml:mo> <mml:mo stretchy=false>(</mml:mo> <mml:msub> <mml:mi>M</mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:mo>×<!-- × --></mml:mo> <mml:msub> <mml:mi>M</mml:mi> <mml:mn>3</mml:mn> </mml:msub> <mml:mo stretchy=false>)</mml:mo> <mml:mo>≥<!-- ≥ --></mml:mo> <mml:mn>4</mml:mn> </mml:mrow> <mml:annotation encoding=application/x-tex>operatorname {cat}(M_2times M_3)ge 4</mml:annotation> </mml:semantics> </mml:math> </inline-formula>." @default.
• W4200403493 created "2021-12-31" @default.
• W4200403493 creator A5084737845 @default.
• W4200403493 date "2022-02-15" @default.
• W4200403493 modified "2023-09-26" @default.
• W4200403493 title "On the LS-category of product of Iwase’s manifolds" @default.
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• W4200403493 doi "https://doi.org/10.1090/proc/15823" @default.
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