2014/09/05 by Andrew Donald · 2 citations
Mathematics · #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #Advanced Combinatorial Mathematics
paper · pdf · doi:10.1090/s0002-9947-2014-06174-6
Using an obstruction based on Donaldson’s theorem on the intersection forms of definite 4-manifolds, we determine which connected sums of lens spaces 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> . We also find constraints on the Seifert invariants of Seifert 3-manifolds 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> when either the base orbifold is non-orientable or the first Betti number is odd. In addition, we construct some new embeddings and use these, along with the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="d"> <mml:semantics> <mml:mi>d</mml:mi> <mml:annotation encoding="application/x-tex">d</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="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> invariants, to examine the question of when the double branched cover of a 3 or 4 strand pretzel link embeds.