Characteristic classes of transversely homogeneous foliations

Abstract

The foliations studied in this paper have transverse geometry modeled on a homogeneous space <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G slash upper H"> <mml:semantics> <mml:mrow> <mml:mi>G</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mi>H</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">G/H</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with transition functions given by the left action of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. It is shown that the characteristic classes for such a foliation are determined by invariants of a certain flat bundle. This is used to prove that when <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is semisimple, the characteristic classes are rigid under smooth deformations, extending work of Brooks, Goldman and Heitsch.