2009/07/16 by J. Hillman, Josh Hillman · 2 citations
Mathematics · Computer Science · #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #Topological and Geometric Data Analysis
paper · pdf · doi:10.1090/s0002-9939-09-09985-7
Constraints on the Seifert invariants of orientable 3-manifolds <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> which admit fixed-point free <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S Superscript 1"> <mml:semantics> <mml:msup> <mml:mi>S</mml:mi> <mml:mn>1</mml:mn> </mml:msup> <mml:annotation encoding="application/x-tex">S1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -actions and embed in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper R Superscript 4"> <mml:semantics> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> <mml:mn>4</mml:mn> </mml:msup> <mml:annotation encoding="application/x-tex">\mathbb R4</mml:annotation> </mml:semantics> </mml:math> </inline-formula> are given. In particular, the generalized Euler invariant of the orbit fibration is determined up to sign by the base orbifold <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper B"> <mml:semantics> <mml:mi>B</mml:mi> <mml:annotation encoding="application/x-tex">B</mml:annotation> </mml:semantics> </mml:math> </inline-formula> unless <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H 1 left-parenthesis upper M semicolon double-struck upper Z right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>H</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>M</mml:mi> <mml:mo>;</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">Z</mml:mi> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">H1(M;\mathbb Z)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is torsion free, in which case it can take at most one nonzero value (up to sign). An <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper H squared times double-struck upper E Superscript 1"> <mml:semantics> <mml:mrow> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">H</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> </mml:msup> <mml:mo> × </mml:mo> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">E</mml:mi> </mml:mrow> <mml:mn>1</mml:mn> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbb H2× \mathbb E1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -manifold <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> with base orbifold <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper B equals upper S squared left-parenthesis alpha 1 comma ellipsis comma alpha Subscript r Baseline right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>B</mml:mi> <mml:mo>=</mml:mo> <mml:msup> <mml:mi>S</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mo stretchy="false">(</mml:mo> <mml:msub> <mml:mi> α </mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>,</mml:mo> <mml:mo> … </mml:mo> <mml:mo>,</mml:mo> <mml:msub> <mml:mi> α </mml:mi> <mml:mi>r</mml:mi> </mml:msub> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">B=S2(α 1,… ,α r)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> where all cone point orders are odd embeds in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper R Superscript 4"> <mml:semantics> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> <mml:mn>4</mml:mn> </mml:msup> <mml:annotation encoding="application/x-tex">\mathbb R4</mml:annotation> </mml:semantics> </mml:math> </inline-formula> if and only if its Seifert data <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S"> <mml:semantics> <mml:mi>S</mml:mi> <mml:annotation encoding="application/x-tex">S</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is skew-symmetric.