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• W1854540643 abstract "Ganea conjectured that for any finite CW complex <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper X> <mml:semantics> <mml:mi>X</mml:mi> <mml:annotation encoding=application/x-tex>X</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and any <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=k greater-than 0> <mml:semantics> <mml:mrow> <mml:mi>k</mml:mi> <mml:mo>></mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding=application/x-tex>k>0</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=c a t left-parenthesis upper X times upper S Superscript k Baseline right-parenthesis equals c a t left-parenthesis upper X right-parenthesis plus 1> <mml:semantics> <mml:mrow> <mml:mi>cat</mml:mi> <mml:mo>⁡<!-- ⁡ --></mml:mo> <mml:mo stretchy=false>(</mml:mo> <mml:mi>X</mml:mi> <mml:mo>×<!-- × --></mml:mo> <mml:msup> <mml:mi>S</mml:mi> <mml:mi>k</mml:mi> </mml:msup> <mml:mo stretchy=false>)</mml:mo> <mml:mo>=</mml:mo> <mml:mi>cat</mml:mi> <mml:mo>⁡<!-- ⁡ --></mml:mo> <mml:mo stretchy=false>(</mml:mo> <mml:mi>X</mml:mi> <mml:mo stretchy=false>)</mml:mo> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding=application/x-tex>operatorname {cat}(Xtimes S^k) =operatorname {cat}(X) + 1</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. In this paper we prove two special cases of this conjecture. The main result is the following. Let <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper X> <mml:semantics> <mml:mi>X</mml:mi> <mml:annotation encoding=application/x-tex>X</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=left-parenthesis p minus 1 right-parenthesis> <mml:semantics> <mml:mrow> <mml:mo stretchy=false>(</mml:mo> <mml:mi>p</mml:mi> <mml:mo>−<!-- − --></mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>(p-1)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-connected <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=n> <mml:semantics> <mml:mi>n</mml:mi> <mml:annotation encoding=application/x-tex>n</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-dimensional CW complex (not necessarily finite). We show that if <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=c a t left-parenthesis upper X right-parenthesis equals left floor StartFraction n Over p EndFraction right floor plus 1> <mml:semantics> <mml:mrow> <mml:mi>cat</mml:mi> <mml:mo>⁡<!-- ⁡ --></mml:mo> <mml:mo stretchy=false>(</mml:mo> <mml:mi>X</mml:mi> <mml:mo stretchy=false>)</mml:mo> <mml:mo>=</mml:mo> <mml:mrow> <mml:mo>⌊</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mfrac> <mml:mi>n</mml:mi> <mml:mi>p</mml:mi> </mml:mfrac> </mml:mrow> <mml:mo>⌋</mml:mo> </mml:mrow> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding=application/x-tex>operatorname {cat}(X) = left lfloor {n over p} right rfloor + 1</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=n not-identical-to minus 1 mod p> <mml:semantics> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>≢</mml:mo> <mml:mo>−<!-- − --></mml:mo> <mml:mn>1</mml:mn> <mml:mi>mod</mml:mi> <mml:mo>⁡<!-- ⁡ --></mml:mo> <mml:mi>p</mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>nnot equiv -1 operatorname {mod} p</mml:annotation> </mml:semantics> </mml:math> </inline-formula> (which implies <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=p greater-than 1> <mml:semantics> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>></mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding=application/x-tex>p>1</mml:annotation> </mml:semantics> </mml:math> </inline-formula>), then <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=c a t left-parenthesis upper X times upper S Superscript k Baseline right-parenthesis equals c a t left-parenthesis upper X right-parenthesis plus 1> <mml:semantics> <mml:mrow> <mml:mi>cat</mml:mi> <mml:mo>⁡<!-- ⁡ --></mml:mo> <mml:mo stretchy=false>(</mml:mo> <mml:mi>X</mml:mi> <mml:mo>×<!-- × --></mml:mo> <mml:msup> <mml:mi>S</mml:mi> <mml:mi>k</mml:mi> </mml:msup> <mml:mo stretchy=false>)</mml:mo> <mml:mo>=</mml:mo> <mml:mi>cat</mml:mi> <mml:mo>⁡<!-- ⁡ --></mml:mo> <mml:mo stretchy=false>(</mml:mo> <mml:mi>X</mml:mi> <mml:mo stretchy=false>)</mml:mo> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding=application/x-tex>operatorname {cat}(Xtimes S^k) =operatorname {cat}(X) +1</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. This is proved by showing that <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=w c a t left-parenthesis upper X times upper S Superscript k Baseline right-parenthesis equals w c a t left-parenthesis upper X right-parenthesis plus 1> <mml:semantics> <mml:mrow> <mml:mi>wcat</mml:mi> <mml:mo>⁡<!-- ⁡ --></mml:mo> <mml:mo stretchy=false>(</mml:mo> <mml:mi>X</mml:mi> <mml:mo>×<!-- × --></mml:mo> <mml:msup> <mml:mi>S</mml:mi> <mml:mi>k</mml:mi> </mml:msup> <mml:mo stretchy=false>)</mml:mo> <mml:mo>=</mml:mo> <mml:mi>wcat</mml:mi> <mml:mo>⁡<!-- ⁡ --></mml:mo> <mml:mo stretchy=false>(</mml:mo> <mml:mi>X</mml:mi> <mml:mo stretchy=false>)</mml:mo> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding=application/x-tex>operatorname {wcat}(Xtimes S^k) =operatorname {wcat}(X) + 1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in a much larger range, and then showing that under the conditions imposed, <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=c a t left-parenthesis upper X right-parenthesis equals w c a t left-parenthesis upper X right-parenthesis> <mml:semantics> <mml:mrow> <mml:mi>cat</mml:mi> <mml:mo>⁡<!-- ⁡ --></mml:mo> <mml:mo stretchy=false>(</mml:mo> <mml:mi>X</mml:mi> <mml:mo stretchy=false>)</mml:mo> <mml:mo>=</mml:mo> <mml:mi>wcat</mml:mi> <mml:mo>⁡<!-- ⁡ --></mml:mo> <mml:mo stretchy=false>(</mml:mo> <mml:mi>X</mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>operatorname {cat}(X)=operatorname {wcat}(X)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. The second special case is an extension of Singhof’s earlier result for manifolds." @default.
• W1854540643 created "2016-06-24" @default.
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• W1854540643 date "1999-09-17" @default.
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• W1854540643 title "Two special cases of Ganea’s conjecture" @default.
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