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• W2106417039 abstract "We prove that if <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper M> <mml:semantics> <mml:mi>M</mml:mi> <mml:annotation encoding=application/x-tex>M</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a CW-complex and <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper M Superscript 1> <mml:semantics> <mml:msup> <mml:mi>M</mml:mi> <mml:mn>1</mml:mn> </mml:msup> <mml:annotation encoding=application/x-tex>M^1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is its 1-skeleton, then the crossed module <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=normal upper Pi 2 left-parenthesis upper M comma upper M Superscript 1 Baseline right-parenthesis> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi mathvariant=normal>Π<!-- Π --></mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:mo stretchy=false>(</mml:mo> <mml:mi>M</mml:mi> <mml:mo>,</mml:mo> <mml:msup> <mml:mi>M</mml:mi> <mml:mn>1</mml:mn> </mml:msup> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>Pi _2(M,M^1)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> depends only on the homotopy type of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper M> <mml:semantics> <mml:mi>M</mml:mi> <mml:annotation encoding=application/x-tex>M</mml:annotation> </mml:semantics> </mml:math> </inline-formula> as a space, up to free products, in the category of crossed modules, with <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=normal upper Pi 2 left-parenthesis upper D squared comma upper S Superscript 1 Baseline right-parenthesis> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi mathvariant=normal>Π<!-- Π --></mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:mo stretchy=false>(</mml:mo> <mml:msup> <mml:mi>D</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mo>,</mml:mo> <mml:msup> <mml:mi>S</mml:mi> <mml:mn>1</mml:mn> </mml:msup> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>Pi _2(D^2,S^1)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. From this it follows that if <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=script upper G> <mml:semantics> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi class=MJX-tex-caligraphic mathvariant=script>G</mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>mathcal {G}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a finite crossed module and <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper M> <mml:semantics> <mml:mi>M</mml:mi> <mml:annotation encoding=application/x-tex>M</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is finite, then the number of crossed module morphisms <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=normal upper Pi 2 left-parenthesis upper M comma upper M Superscript 1 Baseline right-parenthesis right-arrow script upper G> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi mathvariant=normal>Π<!-- Π --></mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:mo stretchy=false>(</mml:mo> <mml:mi>M</mml:mi> <mml:mo>,</mml:mo> <mml:msup> <mml:mi>M</mml:mi> <mml:mn>1</mml:mn> </mml:msup> <mml:mo stretchy=false>)</mml:mo> <mml:mo stretchy=false>→<!-- → --></mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi class=MJX-tex-caligraphic mathvariant=script>G</mml:mi> </mml:mrow> </mml:mrow> <mml:annotation encoding=application/x-tex>Pi _2(M,M^1) to mathcal {G}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> can be re-scaled to a homotopy invariant <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper I Subscript script upper G Baseline left-parenthesis upper M right-parenthesis> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>I</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi class=MJX-tex-caligraphic mathvariant=script>G</mml:mi> </mml:mrow> </mml:mrow> </mml:msub> <mml:mo stretchy=false>(</mml:mo> <mml:mi>M</mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>I_{mathcal {G}}(M)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, depending only on the algebraic 2-type of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper M> <mml:semantics> <mml:mi>M</mml:mi> <mml:annotation encoding=application/x-tex>M</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. We describe an algorithm for calculating <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=pi 2 left-parenthesis upper M comma upper M Superscript left-parenthesis 1 right-parenthesis Baseline right-parenthesis> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>π<!-- π --></mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:mo stretchy=false>(</mml:mo> <mml:mi>M</mml:mi> <mml:mo>,</mml:mo> <mml:msup> <mml:mi>M</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo stretchy=false>(</mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> </mml:msup> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>pi _2(M,M^{(1)})</mml:annotation> </mml:semantics> </mml:math> </inline-formula> as a crossed module over <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=pi 1 left-parenthesis upper M Superscript left-parenthesis 1 right-parenthesis Baseline right-parenthesis> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>π<!-- π --></mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo stretchy=false>(</mml:mo> <mml:msup> <mml:mi>M</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo stretchy=false>(</mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> </mml:msup> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>pi _1(M^{(1)})</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, in the case when <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper M> <mml:semantics> <mml:mi>M</mml:mi> <mml:annotation encoding=application/x-tex>M</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is the complement of a knotted surface <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=normal upper Sigma> <mml:semantics> <mml:mi mathvariant=normal>Σ<!-- Σ --></mml:mi> <mml:annotation encoding=application/x-tex>Sigma</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper S Superscript 4> <mml:semantics> <mml:msup> <mml:mi>S</mml:mi> <mml:mn>4</mml:mn> </mml:msup> <mml:annotation encoding=application/x-tex>S^4</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 Superscript left-parenthesis 1 right-parenthesis> <mml:semantics> <mml:msup> <mml:mi>M</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo stretchy=false>(</mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> </mml:msup> <mml:annotation encoding=application/x-tex>M^{(1)}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is the handlebody of a handle decomposition of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper M> <mml:semantics> <mml:mi>M</mml:mi> <mml:annotation encoding=application/x-tex>M</mml:annotation> </mml:semantics> </mml:math> </inline-formula> made from its <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=0> <mml:semantics> <mml:mn>0</mml:mn> <mml:annotation encoding=application/x-tex>0</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=1> <mml:semantics> <mml:mn>1</mml:mn> <mml:annotation encoding=application/x-tex>1</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-handles. Here, <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=normal upper Sigma> <mml:semantics> <mml:mi mathvariant=normal>Σ<!-- Σ --></mml:mi> <mml:annotation encoding=application/x-tex>Sigma</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is presented by a knot with bands. This in particular gives us a geometric method for calculating the algebraic 2-type of the complement of a knotted surface from a hyperbolic splitting of it. We prove in addition that the invariant <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper I Subscript script upper G> <mml:semantics> <mml:msub> <mml:mi>I</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi class=MJX-tex-caligraphic mathvariant=script>G</mml:mi> </mml:mrow> </mml:mrow> </mml:msub> <mml:annotation encoding=application/x-tex>I_{mathcal {G}}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> yields a non-trivial invariant of knotted surfaces in <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper S Superscript 4> <mml:semantics> <mml:msup> <mml:mi>S</mml:mi> <mml:mn>4</mml:mn> </mml:msup> <mml:annotation encoding=application/x-tex>S^4</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with good properties with regard to explicit calculations." @default.
• W2106417039 created "2016-06-24" @default.
• W2106417039 creator A5046119426 @default.
• W2106417039 date "2009-04-03" @default.
• W2106417039 modified "2023-09-23" @default.
• W2106417039 title "The fundamental crossed module of the complement of a knotted surface" @default.
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