On homotopy groups of spaces of embeddings of an arc or a circle: the Dax invariant

Abstract

We compute in many classes of examples the first potentially interesting homotopy group of the space of embeddings of either an arc or a circle into a 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> of dimension <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="d greater-than-or-equal-to 4"> <mml:semantics> <mml:mrow> <mml:mi>d</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>4</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">d\geq 4</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. In particular, 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 simply connected 4-manifold the fundamental group of both of these embedding spaces is isomorphic to the second homology group 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>, answering a question posed by Arone and Szymik. The case <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="d equals 3"> <mml:semantics> <mml:mrow> <mml:mi>d</mml:mi> <mml:mo>=</mml:mo> <mml:mn>3</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">d=3</mml:annotation> </mml:semantics> </mml:math> </inline-formula> gives isotopy invariants of knots in a 3-manifold that are universal of Vassiliev type <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="less-than-or-equal-to 1"> <mml:semantics> <mml:mrow> <mml:mo>≤</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">\leq 1</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, and reduce to Schneiderman’s concordance invariant.

Locations

• arXiv (Cornell University) - View - PDF
• Transactions of the American Mathematical Society - View