Integral invariants of {3}-manifolds

Raoul Bott, Alberto S. Cattaneo
1998 Journal of differential geometry  
This note describes an invariant of rational homology 3-spheres in terms of configuration space integrals which in some sense lies between the invariants of Axelrod and Singer and those of Kontsevich. Abstract. This note describes an invariant of rational homology 3-spheres in terms of configuration space integrals which in some sense lies between the invariants of Axelrod and Singer [2] and those of Kontsevich [9]. where I Γ (K) is a configuration space integral which is corrected by an
more » ... us term which is a multiple of the self-linking of K. Similarly we now obtain invariants of oriented homology 3-spheres, one for every connected cocycle Γ, of the form so that A Θ (M) is seen to play the role of the self-linking integral in knot theory. Although the invariants of [2] and [5], as well as the ones described here, are all spin-offs from Witten's [11] original Chern-Simons invariants for homology 3-spheres, it seems to us that, from a purely mathematical point of view, they have now, in retrospect, even older antecedents. These are the "iterated integrals" of Chen, or-even olderthe Adams constructions for the loop-space of a space. Quite generally, the principle of these constructions is to describe the cohomology of a function-space F = Map(X, Y ) in terms of the various evaluation maps: When we are dealing with corresponding spaces of imbeddings, or diffeomorphisms, then the configuration spaces enter the discussion quite naturally, and give rise to new invariants of the type we have been discussing. Review of characteristic classes of SO(n) Consider an oriented vector bundle E with odd fiber dimension, n = 2k +1, over a base space M. Also let S(E) denote the associated sphere bundle to E, which we may consider to be the space of rays in E; or, if E is given a Riemannian structure, as the unit sphere bundle of E. In any case S(E) has even fiber dimension 2k over M, and this together with the orientability of E allows one to specify a canonical integral generator of the rational cohomology of S(E) as a module over H * (M). Namely, we consider the "tangent bundle along the fiber," T F S(E), of S(E). This, being an even dimensional oriented bundle, has a canonical Euler class: e = e(T F S(E)) ∈ H 2k (S(E)), (2.1) which restricts to twice the generator of H 2k (S 2k ) on each fiber, because the Euler number of S 2k is 2. But then it follows from general principles that e generates H * (S(E)) over H * (M) over the rationals. Concerning the generator e we have the following lemma, which in some sense explains the Chern-Simons term in our subsequent construction. Lemma 2.1. Let π * denote integration along the fiber in the bundle S(E) over M. Then π * e 3 = 2 p k (E), (2.2)
doi:10.4310/jdg/1214460608 fatcat:5fkhpdqyobckdo5ueahn5ez63q