2020/01/29 by Ahmad Issa, Duncan McCoy · 1 citation
Mathematics · #Geometric and Algebraic Topology #Finite Group Theory Research #Homotopy and Cohomology in Algebraic Topology
paper · doi:10.1090/tran/8095
Using an obstruction based on Donaldson’s theorem, we derive strong restrictions on when a Seifert fibered space <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper Y equals upper F left-parenthesis e semicolon StartFraction p 1 Over q 1 EndFraction comma ellipsis comma StartFraction p Subscript k Baseline Over q Subscript k Baseline EndFraction right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>Y</mml:mi> <mml:mo>=</mml:mo> <mml:mi>F</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>e</mml:mi> <mml:mo>;</mml:mo> <mml:mfrac> <mml:msub> <mml:mi>p</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:msub> <mml:mi>q</mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:mfrac> <mml:mo>,</mml:mo> <mml:mo>…</mml:mo> <mml:mo>,</mml:mo> <mml:mfrac> <mml:msub> <mml:mi>p</mml:mi> <mml:mi>k</mml:mi> </mml:msub> <mml:msub> <mml:mi>q</mml:mi> <mml:mi>k</mml:mi> </mml:msub> </mml:mfrac> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">Y = F(e; \frac p1q1, … , \frac pkqk)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> over an orientable base surface <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper F"> <mml:semantics> <mml:mi>F</mml:mi> <mml:annotation encoding="application/x-tex">F</mml:annotation> </mml:semantics> </mml:math> </inline-formula> can smoothly embed 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">S4</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. This allows us to classify precisely when <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper Y"> <mml:semantics> <mml:mi>Y</mml:mi> <mml:annotation encoding="application/x-tex">Y</mml:annotation> </mml:semantics> </mml:math> </inline-formula> smoothly embeds provided <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="e greater-than k slash 2"> <mml:semantics> <mml:mrow> <mml:mi>e</mml:mi> <mml:mo>></mml:mo> <mml:mi>k</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mn>2</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">e > k/2</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, where <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="e"> <mml:semantics> <mml:mi>e</mml:mi> <mml:annotation encoding="application/x-tex">e</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is the normalized central weight and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="k"> <mml:semantics> <mml:mi>k</mml:mi> <mml:annotation encoding="application/x-tex">k</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is the number of singular fibers. Based on these results and an analysis of the Neumann-Siebenmann invariant <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="mu overbar"> <mml:semantics> <mml:mover> <mml:mi>μ</mml:mi> <mml:mo accent="false">¯</mml:mo> </mml:mover> <mml:annotation encoding="application/x-tex"> μ </mml:annotation> </mml:semantics> </mml:math> </inline-formula>, we make some conjectures concerning Seifert fibered spaces which embed 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">S4</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Finally, we also provide some applications to doubly slice Montesinos links, including a classification of the smoothly doubly slice odd pretzel knots up to mutation.