On Kontsevich’s Characteristic Classes for Smooth 5- and 7-dimensional Homology Sphere Bundles

Kontsevich constructed universal characteristic classes of smooth bundles with fiber a framed odd-dimensional homology sphere, which is known in the 3-dimensional case that they are universal among finite type invariants. The purpose of the present paper is twofold. First, we obtain a bordism invariant of smooth unframed bundles with fiber a 5dimensional homology sphere as a sum of the simplest Kontsevich class and the second signature defect. Second, we introduce the notion of clasper bundles. By using clasper bundles, we show that Kontsevich’s universal characteristic classes are highly non-trivial in the case of fiber dimension 7. CONTENTS