2014/09/04 by Giovanni Bazzoni, Marisa Fernández, Vicente Muñoz · 4 citations
Mathematics · #Geometry and complex manifolds #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology
paper · pdf · doi:10.1090/s0002-9947-2014-06361-7
We study the formality of the mapping torus of an orientation-preserving diffeomorphism of a manifold. In particular, we give conditions under which a mapping torus has a non-zero Massey product. As an application we prove that there are non-formal compact co-symplectic manifolds of dimension <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="m"> <mml:semantics> <mml:mi>m</mml:mi> <mml:annotation encoding="application/x-tex">m</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and with first Betti number <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="b"> <mml:semantics> <mml:mi>b</mml:mi> <mml:annotation encoding="application/x-tex">b</mml:annotation> </mml:semantics> </mml:math> </inline-formula> if and only if <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="m equals 3"> <mml:semantics> <mml:mrow> <mml:mi>m</mml:mi> <mml:mo>=</mml:mo> <mml:mn>3</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">m=3</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="b greater-than-or-equal-to 2"> <mml:semantics> <mml:mrow> <mml:mi>b</mml:mi> <mml:mo> ≥ </mml:mo> <mml:mn>2</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">b ≥ 2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , or <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="m greater-than-or-equal-to 5"> <mml:semantics> <mml:mrow> <mml:mi>m</mml:mi> <mml:mo> ≥ </mml:mo> <mml:mn>5</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">m ≥ 5</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="b greater-than-or-equal-to 1"> <mml:semantics> <mml:mrow> <mml:mi>b</mml:mi> <mml:mo> ≥ </mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">b ≥ 1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . Explicit examples for each one of these cases are given.