Signature Operator Research Articles

Structures Riemanniennes L p et K-Homologie

We construct analytically the signature operator for a new family of topological manifolds. This family contains the quasi-conformal manifolds and the topological manifolds modeled on germs of homeomorphisms of R^n possessing a derivative which is in L^p, with p > n(n+1)/2. We obtain an unbounded Fredholm module which defines a class in the K-homology of the manifold, the Chern character of which is the Hirzebruch polynomial in the Pontrjagin classes of the manifold. This generalizes previous works of N. Teleman for Lipschitz manifolds and of A. Connes, N. Teleman and D. Sullivan for quasi-conformal manifolds of even dimension.

Higher eta invariants and the Novikov conjecture on manifolds with boundary

Let ( N,g) be a closed Riemannian manifold of dimension 2 m-1 and Γ → N ˜ → N be a Galois covering of N. We assume that Γ is of polynomial growth and that Δ N ˜ is L 2-invertible in degree m. By employing spectral sections which are symmetric with respect to the *-Hodge operator, we define the higher eta invariant associated to the signature operator on N ˜ , thus extending previous work of Lott. If π 1 ( M ) → M ˜ → M is the universal cover of a compact orientable even-dimensional manifold with boundary ( ∂M = N) then, under the above invertibility assumption on Δ ∂ M ˜ , we define a canonical Atiyah-Patodi-Singer signature-index class, in K 0 ( C r * (Γ)). Employing the higher APS index theory developed in [4] we express the Chern character of this index class in terms of a local integral and of the higher eta invariant defined above. We apply these results to the problem of the existence and homotopy invariance of higher signatures on manifolds with boundary.

Periodic cyclic cohomology Chern character for pseudomanifolds with one singular stratum

We compute the periodic cyclic cohomology Chern character of an admissible pseudomanifold X † X^{\dagger } with one singular stratum. As a corollary, we obtain the index theorem and spectral flow for signature operators.

A signature formula for manifolds with corners of codimension two

We present a signature formula for compact 4 k-manifolds with corners of codimension two which generalizes the formula of Atiyah et al. for manifolds with boundary. The formula expresses the signature as a sum of three terms, the usual Hirzebruch term given as the integral of an L-class, a second term consisting of the sum of the eta invariants of the induced signature operators on the boundary hypersurfaces with Atiyah-Patodi-Singer boundary condition (augmented by the natural Lagrangian subspace, in the corner null space, associated to the hypersurface) and a third “corner” contribution which is the phase of the determinant of a matrix arising from the comparison of the Lagrangians from the different hypersurfaces ing at the corners. To prove the formula, we pass to a complete metric, apply the Atiyah-Patodi-Singer formula for the manifold with the corners “rounded” and then apply the results of our previous work [11] describing the limiting behaviour of the eta invariant under analytic surgery in terms of the b-eta invariants of the final manifold(s) with boundary and the eta invariant of a reduced, one-dimensional, problem. The corner term is closely related to the signature defect discovered by Wall [25] in his formula for nonadditivity of the signature. We also discuss some product formulae for the b-eta invariant.

C*-Algebras and Controlled Topology

This paper is an attempt to explain some aspects of the relationship between the K-theory of C-algebras, on the one hand, and the categories of modules that have been developed to systematize the algebraic aspects of controlled topology, on the other. It has recently become apparent that there is a substantial conceptual overlap between the two theories, and this allows both the recognition of common techniques, and the possibility of new methods in one theory suggested by those of the other. In this first part we will concentrate on defining the C-algebras associated to various kinds of controlled structure and giving methods whereby their K-theory groups may be calculated in a number of cases. From a ‘revisionist’ perspective, this study originates from an attempt to relate two approaches to the Novikov conjecture. The Novikov conjecture states that a certain assembly map is injective. The process of assembly can be thought of as the formation of a ‘generalized signature’ [38], and therefore to understand the connection between different approaches to the conjecture is to understand the connection between different definitions of the ‘generalized signature’. Now, broadly speaking, there are two approaches to the Novikov conjecture in the literature. One approach considers the original assembly map of Wall in L-theory, and attempts to prove it to be injective by investigating homological properties of the L-theory groups (which properties themselves may be derived algebraically, or geometrically, by relating them to surgery problems). The other approach proceeds via analysis, considering the assembly map to be the formation of a generalized index of the Atiyah-Singer signature operator [3]. This approach ultimately leads to the consideration of assembly on the K-theory of C-algebras. Nevertheless it can be shown that the injectivity of assembly on the C-algebra level implies (modulo 2-torsion) the injectivity of assembly on the L-theory level. All this is explained in the paper of Rosenberg [37], to which we also refer for an extensive bibliography on the Novikov conjecture. To proceed with the background to this paper. In the late seventies and eighties it occurred both to the topologists and independently to the analysts that a more flexible generalized signature theory might be developed if one neglected the group structure of the fundamental group π in question and considered only its “large scale” or “coarse” structure induced by some translation invariant metric; up to “large scale equivalence” the choice of such a metric is irrelevant. Some references are [26, 29, 11, 12, 14, 16, 9, 33, 34, 35, 36, 41]. Since the two theories were based on the same idea, it was inevitable that they would eventually come into interaction, but this did not happen for some while. It was the insight of Shmuel Weinberger, and especially his note [40] relating the index theory of [36] to Novikov’s theorem on the topological invariance of the rational Pontrjagin classes, which provoked the discussions among the present authors which eventually led to the writing of this paper. This paper is intended to be foundational, setting down some of the language in which one can talk about the relationship between analysis and controlled topology. Many of its ideas have already been worked out in the special case of bounded control in the work of the first and third authors and G. Yu [18, 20, 19]. But it seems that there is much to be gained by considering more general kinds of control, more general coefficients, “spacification” of the theory and so on, and it

The spectral flow of the odd signature operator and higher Massey products

We show how to compute the spectral flow of the odd signature operator $\pm *d_{a_t}-d_{a_t}*$ along an analytic path of flat connections $a_t$ on a bundle over a closed odd-dimensional manifold in terms of Massey products in the DGLA of bundle-valued differential forms. To obtain this information, we set up a sequence of cochain complexes $\{\calg^*_n,\delta_n\}$, for $n=0,1,2,\ldots$ and Hermitian forms $$Q_n:\calg_n\times\calg_n\ra \bbbC$$ whose signatures determine the spectral flow through $t=0$. The complexes and Hermitian forms are constructed using Massey products.

Analytic deformations of equivariant genera, universal symmetry blocks and Witten's rigidity

This paper is an extension of [9], where we have established a fixed-point formula for the formal t-deformations ind,(D, M) of the G-index ind(D, M) (G being a compact Lie group) of a basic differential operator D on a closed G-manifold M. The most important examples of D are the Signature, Dirac, and Euler-Todd operators. This deformation ind,(D, M) takes value in the ring %!(G)[ [t]], g’(G) denoting the algebra of central functions on G, @-spanned by the characters of the complex G-representations. The deformation is produced with the help of a stable exponential operation 4: K,( -) + @ @ KG( -)[ [t]] in the equivariant complex K-theory. In fact, these operations are in one-to-one correspondence with the elements cpt(u) E C [u, u- ‘1 [[t]], cpO(u) being invertible in C [u, u- ‘1. In the text we can also use notation Q(M) for ind,(D, M). One can view ind,(D, M) as a formal sum C, ~ o Xjt’, where xj is a G-index of the operator D twisted by certain tensor fields on M. The type of the twist is prescribed by q,(u), j, and dim M (cf. (1.3) and (1.6) from Section 1).

The resolvent expansion for the signature operator on a manifold with a conic singular stratum

The resolvent expansion for the signature operator on a manifold with a conic singular stratum

Geometric invariants for Seifert fibred 3-manifolds

In this paper, we obtain a formula for the η \eta -invariant of the signature operator for some circle bundles over Riemannian 2 2 -orbifolds. We then apply it to Seifert fibred 3 3 -manifolds endowed with one of the six Seifert geometries. By using a relation between the Chern-Simons invariant and the η \eta -invariant, we also derive some elementary formulae for the Chern-Simons invariant of these manifolds. As applications, we show that some families of these manifolds cannot be conformally immersed into the Euclidean space E 4 {{\mathbf {E}}^4} .

Spectral flow, eta invariants, and von Neumann algebras

In this paper, we prove that the spectral flow of a path of tangential signature operators on a closed odd dimensional manifold parametrized by unitary representations of the fundamental group is a homotopy invariant of the manifold and the path in the space of unitary representations, for a large class of fundamental groups with the help of results due to Weinberger ( Proc. Nat. Acad. Sci. U.S.A. 85 (1988), 5362–5363). We also prove that an eta invariant for normal covering spaces introduced by Cheeger and Gromov ( J. Differential Geom. 21 (1985), 1–34) and defined using the von Neumann trace, is a homotopy invariant for Bieberbach groups. Finally, we compute this invariant for closed locally symmetric spaces, and obtain the surprising result that the eta invariant for such spaces is a differential invariant.

Seifert-fibred homology 3-spheres,V-surfaces and the Floer index

Let Σ be a Seifert-fibred homology 3-sphere. We are interested in the chain complex for the Floer homology of Σ. This is generated by the critical points of the ChernSimons functional acting on the moduli space of irreducibleSU(2)-connections modulo gauge-equivalence, i.e. the equivalence classes of flat connections: see [6]. Specifically, we ask the question: given the holonomy ρ of a flat connectionCρ, what is the index ofCρin the chain complex? By definition thisFloer indexis given by the spectral flow of a family of twisted signature operators with coefficients in the adjoint bundle (with fibre su(2)) corresponding to any path of connections joining the trivial connection toCρ. We will calculate this spectral flow by an almost entirely direct method, obtaining a formula in terms of the dimensions of spaces of π1(Σ)- automorphic functions. These dimensions will be evaluated to give the numerical result previously obtained using a different method by Fintushel and Stern: see [5].

A compact imbedding result on Lipschitz manifolds

In [14], Teleman developed a theory of signature operators on Lipschitz manifolds. The theory includes both the classical index operator _D + and "twisted" operators with values in vector bundles. The main result of 1,-14] is Theorem 13.1 which states, in part, that for any closed, oriented, Lipschitz Riemannian manifold M" and for any Lipschitz complex vector bundle ~ over M, there is a natural signature operator _D[; the index of this operator is a Lipschitz invariant of the pair (M, ~). Using this result, Sullivan and Teleman 1-13] were able to show that (except in dimension 4) the index of _D~is actually a topological invariant of(M, 4). The computation of the index of _D~requires three tools: Hodge theory, a compact imbedding result and an excision theorem. The proof of the imbedding result in particular is quite complicated, using ideas from the theory of pseudodifferential operators. The main purpose of the present paper is to provide a simpler proof of the compact imbedding, based on ideas in 1-11], and to show that Hodge theory may be derived easily as a consequence of the imbedding result. (In the smooth case, this approach to Hodge theory may be traced back to Gaffney 15] and Friedrichs I-4].) Our techniques use only elementary facts from functional analysis. (For a concise summary of these see Appendix A of 1,8].) Before proceeding with the rigorous development, we would like to explain briefly the essential new idea in our approach. (The precise definitions of all spaces and operators may be found in the next section.) We denote the Hilbert space of square-integrable differential forms on M by L2(M), and define

Adiabatic limits, nonmultiplicativity of signature, and Leray spectral sequence

We first prove an adiabatic limit formula for the η \eta -invariant of a Dirac operator, generalizing the recent work of J.-M. Bismut and J. Cheeger. An essential part of the proof is the study of the spectrum of the Dirac operator in the adiabatic limit. A new contribution arises in the adiabatic limit formula, in the form of a global term coming from the (asymptotically) very small eigenvalues. We then proceed to show that, for the signature operator, these very small eigenvalues have a purely topological significance. In fact, we show that the Leray spectral sequence can be recast in terms of these very small eigenvalues. This leads to a refined adiabatic limit formula for the signature operator where the global term is identified with a topological invariant, the signature of a certain bilinear form arising from the Leray spectral sequence. As an interesting application, we give intrinsic characterization of the non-multiplicativity of signature.

Homotopy invariance of η-invariants

Intersection homology and results related to the higher signature problem are applied to show that certain combinations of eta-invariants of the signature operator are homotopy invariant in various circumstances.

Local index theorem for signature operators

Local index theorem for signature operators

Group actions and equivariant Lipschitz analysis

The idea that one could prove by analytic methods the topological invariance of characteristic classes was suggested in an influential lecture by Singer. The concrete realization of this, in the case of rational Pontrjagin classes, was the result of works by Sullivan, who showed that topological manifolds have essentially unique Lipschitz structures, and Teleman, who showed that Lipschitz manifolds are the type of object for which one can analyze the signature operator. (A refinement of the same method proves the KO[l/2] orientability of topological bundles.) Much of this can be done for manifolds with group actions. The main implication is that for certain sorts of topological group actions (including all smooth and PL actions) one can construct an signature operator whose class in equivariant if-homology is a topological invariant. This class satisfies the G-signature theorem. The topological method has implications for various precise classification problems (e.g. semifree topological actions on the sphere), but here we shall present only the consequences of these Lipschitzian ideas: (A) Topologically locally linear G-manifolds for odd-order G have KOG[1/2] orientations (Madsen-Rothenberg). For all finite G there are Lipschitz structures, but the signature class is a zero divisor for even-order groups. The same method produces, upon application of the localization theorem, Atiyah-Singer classes for some actions with nonmanifold fixed set (Weinberger, extending earlier joint work with Cappell). (B) For odd-order groups, topological and linear conjugacy of linear representations are equivalent (Hsiang-Pardon, Madsen-Rothenberg). This is a consequence of (A). Alternatively one can return to the original argument of Atiyah-Bott-Milnor for semifree representations. A stronger result is true: the Atiyah-Bott local numbers associated to representations agree for elements that have discrete fixed sets in topologically equivalent representations. This distinguishes many pairs of representations of even-order groups. (C) R Top(G) = R Lip(G) for all finite G, where R Cat is the Grothendieck group of representations up to Cat conjugacy (Weinberger, after inverting 2). This implies that there is no analytic definition of Reidemeister torsion that works well in the Lipschitz setting. The reason is that, following

A short proof of the local Atiyah-Singer index theorem

(Received 24 April 1985) IN THIS paper, we will give a simple proof of the local Atiyah-Singer index theorem first proved by Patodi [9]; in fact, his earlier proof of the Gauss-Bonnet-Chern theorem (Patodi [8]) is quite close to ours. (Perhaps he did not find the proof for Dirac operators given in this paper because he was unaware of the symbol calculus for Clifford algebras.) A paper of Kotake [S] contains a proof of the Riemann-Roth theorem for Riemann surfaces along similar lines, and a recent paper of Bismut [3] is also very closely related. It might be helpful to give a short history of this theorem. As explained in Atiyah et al. [l], all of the common geometric complexes, namely, the twisted Dirac operators, &operators, signature operators and the De Rham complex, are, locally, Dirac operators. We shall refer to all of these operators as Dirac operators, although this is not globally correct on non-spin manifolds. The index theorem for Dirac operators was first proven, at least for Kahler manifolds, by Hirzebruch using cobordism theory. A few years later, McKean and Singer gave their famous formula for the index of the Dirac operator: Index (D) = Str efo2( = Tr erp-e’ - Tr erpcm-)

Axial anomaly and index theorem for Dirac-K�hler fermions

We present a calculation of the axial anomaly for Dirac-Kahler fermions in two and four dimensions applying the procedure developed by Seeley to the signature operator in the twisted complex. The result is equal to the one for the twisted spin complex times 2π/2 (n: number of dimensions) and agrees with the expressions from the index theorem.

The index of signature operators on Lipschitz manifolds

The index of signature operators on Lipschitz manifolds

Combinatorial hodge theory and signature operator

This paper represents an detailed version of our paper [12] presented at the Hawaii Conference on the Geometry of the Laplace Operator, March 1979. In this paper we present the solution to the following problem posed by Singer in [8], w 4, concerning elliptical operators on PL-maifolds: "If M is a PL-manifold, the L-polynomials are still well defined (Thom [13]) from which one can still define the rational Pontrjagin classes. The Hirzebruch signature theorem still holds. Is there an associated elliptic operator (as in the smooth case) whose index is the signature? On what spaces does it operate? Does it have a symbol and where does the symbol lie?"