Invariants Of Homotopy Type Research Articles

Homotopy invariance of foliation Betti numbers

Foliation Betti numbers, introduced by Connes [C I], bear a striking formal similarity to the Betti numbers of a Galois covering space [A]. Both appear in a index theorem for DeRham complexes, and both can be interpreted as the trace on a v o n Neumann algebra applied to projection operators. Dodziuk proved that the covering space Betti numbers are invariants of homotopy type [D]. The purpose of this paper is to prove the corresponding result for foliations: the leafwise Betti numbers of a foliation of a compact manifold which admits an invariant transverse measure are preserved by a measure preserving leafwise homotopy equivilence. Given two compact foliated manifolds with invariant transverse measures and a leaf-wise homotopy equivalence, we construct a v o n Neumann algebra containing the yon Neumann algebras of the individual foliations and we show that there is an involution in this algebra which intertwines the corresponding projections which give rise to the Betti numbers of the two foliations. The assumption that the homotopy equivalence is measure preserving implies that there is a trace on this new algebra which restricts to the traces on the von Neumann algebras of the two foliations. Intertwining implies that the projections have the same traces, and thus that the corresponding Betti numbers are the same. The main technical problems are the construction of a transversely measureable leaf-wise triangulation whose simplicies have volume and diameter bounded away from zero, and the construction of simplicial approximations to the leafwise homotopy equivilence and its inverse. The techniques of I-D] and [H-L I] are then applicable.

Category of loop spaces of open subsets in euclidean space

has been studied in several recent papers (e.g. [l, 2, 5, 6, 91) in situations where the potential function V is singular, in the sense that it is defined and C’ only on an open subset U of Euclidean k-space Rk and not on the whole of Rk. If E = WGT2(R, Rk) is the Sobolev space of T-periodic functions from R to Rk and E(U) is the subset of E of functions which map into U, then the topology of E(U) plays a crucial role in these studies. In particular, the knowledge that the Lusternik-Schnirelmann category of E(U) is infinite, when used in conjunction with min-max methods, implies the existence of infinitely many critical points. Another subset E,,(U), which consists of those functions in E(U) which take the origin to a base point in U, plays a dual role. First of all, in problems concerning the existence of geodesics in U from one subset A of U to another B of U (see e.g. [3]), the category of E,(U) plays a role. Secondly, the general inequality cat E(U) 2 cat E,,(U) (see [6]), when U is connected and simply connected, can be applied to show that cat E(U) = cat E,,(U), thus providing a tool for computing cat E(U). The topological analogue of E(U) is the space of free loops AU = C’[S’, U] of continuous maps from the circle S’ to U and the analogue of E,(U) is the space of based loops RU which consists of maps in AU which take a base point in S’ to a base point in U. AU has the same homotopy type as E(U) and S2U has the same homotopy type as E,,(U). Thus, homotopy type invariants (e.g. category) of E(U) and E,(U) may be studied through AU and SZU, respectively. The following classical theorem of Serre [l l] is useful in this regard and is stated in terms of a general space X.

On the inertia groups of ℎ-cobordant manifolds

It is shown that if M M and N N are h h -cobordant manifolds of dimension at least eight, then their special inertia groups are equal. (G. Brumfiel has shown that special inertia groups are not homotopy type invariants.) Examples are constructed for which the inertia group is an h h -cobordism invariant.

On the Lusternik-Schnirelmann Category of Abstract Groups

Let H be an abstract group. The cohomology groups H'(1l, A) may then be considered for any integer q > 0 and any abelian group A with II as a group of operators. The least integer n such that H'(II, A) = 0 for all A and all q > n is called the dimension of I. If no such integer exists then dim II = 00. For given HI, one can construct a connected aspherical CW-complex K11 with H as fundamental group.' Any two such complexes have the same homotopy type. The least dimension of such a complex Ku is called the geometric dimension of H. It is always infinite if H contains elements of finite order. The (Lusternik-Schnirelmann) category of a topological space is the least integer n such that can be covered by open sets UO, * * *, Un such that Ui is contractible in X. If no such integer exists then cat = oo. If we replace the phrase UE is contractible in X by each closed path in Us is contractible in X we obtain the 1-dimensional category of (notation: cat, X).2 Both cat and cat1 are homotopy type invariants. The category of the aspherical complex Ku (which is independent of the choice the complex K11) is called the category of I. Since K11 is aspherical we have cat Ku = cat1 Ku1, there is therefore no need to define cat IH. The main result of this note is THEOREM 1. For any group HI