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• W2022697446 abstract "We extend the definition of Bockstein basis <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=sigma left-parenthesis upper G right-parenthesis> <mml:semantics> <mml:mrow> <mml:mi>σ<!-- σ --></mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>G</mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>sigma (G)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> to nilpotent groups <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper G> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding=application/x-tex>G</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. A metrizable space <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> is called a <italic>Bockstein space</italic> if <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=dimension Subscript upper G Baseline left-parenthesis upper X right-parenthesis equals sup left-brace right-brace vertical-bar of dimdimH left-parenthesis right-parenthesis upper X vertical-bar element-of element-of upper H of sigma sigma left-parenthesis right-parenthesis upper G> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>dim</mml:mi> <mml:mi>G</mml:mi> </mml:msub> <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:mo movablelimits=true form=prefix>sup</mml:mo> <mml:mo fence=false stretchy=false>{</mml:mo> <mml:msub> <mml:mi>dim</mml:mi> <mml:mi>H</mml:mi> </mml:msub> <mml:mo>⁡<!-- ⁡ --></mml:mo> <mml:mo stretchy=false>(</mml:mo> <mml:mi>X</mml:mi> <mml:mo stretchy=false>)</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo stretchy=false>|</mml:mo> </mml:mrow> <mml:mi>H</mml:mi> <mml:mo>∈<!-- ∈ --></mml:mo> <mml:mi>σ<!-- σ --></mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>G</mml:mi> <mml:mo stretchy=false>)</mml:mo> <mml:mo fence=false stretchy=false>}</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>operatorname {dim}_G(X) = sup {operatorname {dim}_H(X) | Hin sigma (G)}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for all Abelian groups <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper G> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding=application/x-tex>G</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. The Bockstein First Theorem says that all compact spaces are Bockstein spaces. Here are the main results of the paper: <bold>Theorem 0.1.</bold> <italic>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 Bockstein space. If <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper G> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding=application/x-tex>G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is nilpotent, then <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=dimension Subscript upper G Baseline left-parenthesis upper X right-parenthesis less-than-or-equal-to 1> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>dim</mml:mi> <mml:mi>G</mml:mi> </mml:msub> <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 {dim}_G(X) leq 1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> if and only if <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=sup left-brace right-brace vertical-bar of dimdimH left-parenthesis right-parenthesis upper X vertical-bar element-of element-of upper H of sigma sigma left-parenthesis right-parenthesis upper G less-than-or-equal-to 1> <mml:semantics> <mml:mrow> <mml:mo movablelimits=true form=prefix>sup</mml:mo> <mml:mo fence=false stretchy=false>{</mml:mo> <mml:msub> <mml:mi>dim</mml:mi> <mml:mi>H</mml:mi> </mml:msub> <mml:mo>⁡<!-- ⁡ --></mml:mo> <mml:mo stretchy=false>(</mml:mo> <mml:mi>X</mml:mi> <mml:mo stretchy=false>)</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo stretchy=false>|</mml:mo> </mml:mrow> <mml:mi>H</mml:mi> <mml:mo>∈<!-- ∈ --></mml:mo> <mml:mi>σ<!-- σ --></mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>G</mml:mi> <mml:mo stretchy=false>)</mml:mo> <mml:mo fence=false stretchy=false>}</mml:mo> <mml:mo>≤<!-- ≤ --></mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding=application/x-tex>sup {operatorname {dim}_H(X) | Hin sigma (G)}leq 1</mml:annotation> </mml:semantics> </mml:math> </inline-formula>.</italic> <bold>Theorem 0.2.</bold> <italic><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> is a Bockstein space if and only if <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=dimension Subscript bold upper Z Sub Subscript left-parenthesis l right-parenthesis Baseline left-parenthesis upper X right-parenthesis equals dimension Subscript ModifyingAbove upper Z With caret Sub Subscript left-parenthesis l right-parenthesis Baseline left-parenthesis upper X right-parenthesis> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>dim</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msub> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=bold>Z</mml:mi> </mml:mrow> </mml:mrow> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo stretchy=false>(</mml:mo> <mml:mi>l</mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> </mml:msub> </mml:mrow> </mml:msub> <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:msub> <mml:mi>dim</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msub> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mover> <mml:mi>Z</mml:mi> <mml:mo stretchy=false>^<!-- ^ --></mml:mo> </mml:mover> </mml:mrow> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo stretchy=false>(</mml:mo> <mml:mi>l</mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> </mml:msub> </mml:mrow> </mml:msub> <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 {dim}_{{mathbf {Z}}_{(l)}} (X) = operatorname {dim}_{hat {Z}_{(l)}}(X)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for all subsets <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=l> <mml:semantics> <mml:mi>l</mml:mi> <mml:annotation encoding=application/x-tex>l</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of prime numbers.</italic>" @default.
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• W2022697446 date "2009-11-23" @default.
• W2022697446 modified "2023-09-26" @default.
• W2022697446 title "Bockstein theorem for nilpotent groups" @default.
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