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We prove generating series formulae for suitable twisted characteristic classes of symmetric products of a singular complex quasi-projective variety. More concretely, we study homology Hirzebruch classes for motivic coefficients, as well as for complexes of mixed Hodge modules. As a special case, we obtain a generating series formula for the (intersection) homology Hirzebruch classes of symmetric products. In some cases, the latter yields a similar formula for twisted homology L-classes generalizing results of Hirzebruch-Zagier and Moonen. Our methods also apply to the study of Todd classes of (complexes of) coherent sheaves, as well as Chern classes of (complexes of) constructible sheaves, generalizing to arbitrary coefficients results of Moonen and resp. Ohmoto., Paper rewritten to also include generating series formulae for motivic homology Hirzebruch classes, Todd classes of coherent sheaves, as well as Chern classes of constructible sheaves. Some special cases of L-class formulae are also obtained. Moreover, we give a mixed-Hodge-module-free proof for the case of motivic homology Hirzebruch classes

Let M be a locally symmetric irreducible closed manifold of dimension ≥3. A result of Borel [6] combined with Mostow rigidity imply that there exists a finite group G = G ( M ) such that any finite subgroup of Homeo + ( M ) is isomorphic to a subgroup of G. Borel [6] asked if there exist M's with G ( M ) trivial and if the number of conjugacy classes of finite subgroups of Homeo + ( M ) is finite. We answer both questions: (1) For every finite group G there exist M's with G ( M ) = G , and (2) the number of maximal subgroups of Homeo + ( M ) can be either one, countably many or continuum and we determine (at least for dim ⁡ M ≠ 4 ) when each case occurs. Our detailed analysis of (2) also gives a complete characterization of the topological local rigidity and topological strong rigidity (for dim M ≠ 4 ) of proper discontinuous actions of uniform lattices in semisimple Lie groups on the associated symmetric spaces.

Homology Hirzebruch characteristic classes for singular varieties have been recently defined by Brasselet-Schuermann-Yokura as an attempt to unify previously known characteristic class theories for singular spaces (e.g., MacPherson-Chern classes, Baum-Fulton-MacPherson Todd classes, and Goresky-MacPherson L-classes, respectively). In this note we define equivariant analogues of these classes for singular quasi-projective varieties acted upon by a finite group of algebraic automorphisms, and show how these can be used to calculate the homology Hirzebruch classes of global quotient varieties. We also compute the new classes in the context of monodromy problems, e.g., for varieties that fiber equivariantly (in the complex topology) over a connected algebraic manifold. As another application, we discuss Atiyah-Meyer type formulae for twisted Hirzebruch classes of global orbifolds., Comment: v2: updates include a motivic approach, as well as an Atiyah-Meyer formula for global orbifolds, including a defect formula

We study the problem in transformation groups of realizing a manifold simple homotopy equivalent to a component of the fixed point set of a G-manifold as (a component of) the fixed set of another G-manifold equivariantly simple homotopy equivalent to the original one. We show that such replacability of (a component of) the fixed points is very often possible if the normal representation of the fixed point component is twice a complex representation (and a monodromy vanishes). In addition, we discuss for various compact groups some examples displaying topological phenomenona ranging from replacablility to rigidity.

The work of Ray and Singer that introduced analytic torsion, a kind of determinant of the Laplacian operator in topological and holomorphic settings, is naturally generalized in both settings. The couplings are extended in a direct way in the topological setting to general flat bundles and in the holomorphic setting to bundles with (1,1) connections, which, by using the Newlander-Nirenberg theorem, are seen to be the bundles with both holomorphic and antiholomorphic structures. The resulting natural generalizations of Laplacians are not always self-adjoint, and the corresponding generalizations of analytic torsions are thus not always real-valued. The Cheeger-Muller theorem on equivalence in a topological setting of analytic torsion to classical topological torsion generalizes to this complex-valued torsion. On the algebraic side the methods introduced include a notion of torsion associated to a complex equipped with both boundary and coboundary maps. © 2009 Wiley Periodicals, Inc.

We study the behavior of Hodge-theoretic genera under morphisms of complex algebraic varieties. We prove that the additive $\chi_y$-genus which arises in the motivic context satisfies the so-called ``stratified multiplicative property", which shows how to compute the invariant of the source of a proper surjective morphism from its values on various varieties that arise from the singularities of the map. By considering morphisms to a curve, we obtain a Hodge-theoretic analogue of the Riemann-Hurwitz formula. We also study the contribution of monodromy to the $\chi_y$-genus of a smooth projective family, and prove an Atiyah-Meyer type formula for twisted $\chi_y$-genera. This formula measures the deviation from multiplicativity of the $\chi_y$-genus, and expresses the correction terms as higher-genera associated to cohomology classes of the quotient of the total period domain by the action of the monodromy group. By making use of Saito's theory of mixed Hodge modules, we also obtain formulae of Atiyah-Meyer type for the corresponding Hirzebruch characteristic classes., Comment: V2: No major changes in content, but some results are stated in greater generality (original proofs not altered); examples added; corrected a calculation in Example 2.9(2) following a comment of J. Steenbrink; updated references, removed typos, improved exposition; the section on intersection homology genera removed (it will be incorporated in a future paper)

This note studies the behavior of Euler characteristics and of intersection homology Euler characteristics under proper morphisms of algebraic (respectively, analytic) varieties. The methods also yield, for algebraic (respectively, analytic) varieties, formulae comparing these two kinds of Euler characteristics. The main results are direct consequences of the calculus of constructible functions and Grothendieck groups of constructible sheaves. Similar formulae for Hodgetheoretic invariants of algebraic varieties under morphisms were announced by the first and third authors in [5, 14]. c � 2007 Wiley Periodicals, Inc.

The aim of this paper is to study the behavior of Hodge-theoretic (intersection homology) genera and their associated characteristic classes under proper morphisms of complex algebraic varieties. We obtain formulae that relate (parametrized families of) global invariants of a complex algebraic variety X to such invariants of singularities of proper algebraic maps defined on X. Such formulae severely constrain, both topologically and analytically, the singularities of complex maps, even between smooth varieties. Similar results were announced by the first and third author in [13, 32]. © 2007 Wiley Periodicals, Inc.

This paper considers three separate matrices associated to graphs and (each dimension of) cell complexes. It relates all the coefficients of their respective characteristic polynomials to the geometric and combinatorial enumeration of three kinds of subobjects. The matrices are: the mesh matrix for integral d-cycles of Trent, the mesh matrix for integral d-boundaries, and the Kirchhoff matrix, i.e., the combinatorial Laplacian, for integral (d-1)-chains. Relations to Reidemeister-Franz torsion are elucidated and relations to the foundational work of R. Lyons and G. Kalai., 34 pages, Latex:Reorganized to increase comprehensibility

This paper establishes an Atiyah-type characteristic class formula for both the twisted signature and the twisted L-classes of a singular space. The result, in its present form, applies to coefficient systems that satisfy a transversality condition near the singularities. This condition is frequently automatically satisfied, e.g. on supernormal spaces or for local systems arising in certain geometric mapping situations. We obtain applications to stratified maps by combining our formula with the Cappell-Shaneson signature formula.

A perturbative SU(3) Casson invariant \( \Lambda_{SU(3)}(X) \) for integral homology 3-sphere is defined. Besides being fully perturbative, it has the nice properties: (1) \( 4\cdot \Lambda_{SU(3)} \) is an integer. (2) It is preserved under orientation change. (3) A connect sum formula. Explicit calculations of the invariant for 1/k surgery of (2, q) torus knot are presented and compared with Boden-Herald‚s different SU(3) generalization of Casson‚s invariant. For those cases computed, the invariant defined here is a quadratic polynomial in k for k > 0 and a different quadratic polynomial for k < 0.

The theory of spectral flows developed in the series [10, 11, 12], and the present paper has a wide range of applications to important geometric operators on compact manifolds. To present our results on spectral flow and manifold decomposition, the present paper develops a theory of determinant line bundles and infinitedimensional Lagrangians associated to self-adjoint elliptic operators on compact manifolds. The trace-class properties of these infinite Lagrangians established here and the precise uniform estimates relating them to finite Lagrangians are crucial for such a determinant line bundle approach to analytical questions. As an application, we elucidate the Walker’s and other generalizations of Casson’s SU(2) representation theoretic invariant of 3-manifolds in terms of the -invariant of certain Dirac operators. This is carried out by introducing the technique of “canonical perturbations” of singular Lagrangian subvarieties in symplectic geometry. At the end of Part II of this series, we obtained a formulation of the spectral flow of a family of self-adjoint elliptic operators D(u) : L2(E)! L2(E) in terms

This the second part of a three-part investigation of the behavior of certain analytical invariants of manifolds that can be split into the union of two submanifolds. In Part I we studied a splicing construction for low eigenvalues of self-adjoint elliptic operators over such a manifold. Here we go on to study parameter families of such operators and use the previous “static” results in obtaining results on the decomposition of spectral flows. Some of these “dynamic” results are expressed in terms of Maslov indices of Lagrangians. The present treatment is sufficiently general to encompass the difficulties of zero-modes at the ends of the parameter families as well as that of “jumping Lagrangians.” In Part 111, we will compare infinite- and finite-dimensional Lagrangians and determinant line bundles and then introduce “canonical perturbations” of Lagrangian subvarieties of symplectic varieties. We shall then use this information to study invariants of 3-manifolds, including Casson’s invariant. 0 1996

This paper is the first of a three-part investigation into the behavior of analytical invariants of manifolds that can be split into the union of two submanifolds. In this article, we will show how the low eigensolutions of a self-adjoint elliptic operator over such a manifold can be studied by a splicing construction. This construction yields an approximated solution of the operator whenever we have two L2-solutions on both sides and a common limiting value of two extended L2-solutions. In Part 11, the present analytic “Mayer-Vietoris” results on low eigensolutions and further analytic work will be used to obtain a decomposition theorem for spectral flows in terms of Maslov indices of Lagrangians. In Part I11 after comparing infinite- and finite-dimensional Lagrangians and determinant line bundles and then introducing “canonical perturbations” of Lagrangian subvarieties of symplectic varieties, we will study invariants of 3-manifolds, including Casson’s invariant. 0 1996 John Wiley & Sons, Inc.

We obtain a formula for the generating series of (the push-forward under the Hilbert-Chow morphism of) the Hirzebruch homology characteristic classes of the Hilbert schemes of points for a smooth quasi-projective variety of arbitrary pure dimension. This result is based on a geometric construction of a motivic exponentiation generalizing the notion of motivic power structure, as well as on a formula for the generating series of the Hirzebruch homology characteristic classes of symmetric products. We apply the same methods for the calculation of generating series formulae for the Hirzebruch classes of the push-forwards of "virtual motives" of Hilbert schemes of a threefold. As corollaries, we obtain counterparts for the MacPherson (and Aluffi) Chern classes of Hilbert schemes of a smooth quasi-projective variety (resp. for threefolds). For a projective Calabi-Yau threefold, the latter yields a Chern class version of the dimension zero MNOP conjecture., comments are welcome

In this paper we give four definitions of Maslov index and show that they all satisfy the same system of axioms and hence are equivalent to each other. Moreover, relationships of several symplectic and differential geometric, analytic, and topological invariants (including triple Maslov indices, eta invariants, spectral flow and signatures of quadratic forms) to the Maslov index are developed and formulae relating them are given. The broad presentation is designed with a view to applications both in geometry and in analysis. © 1994 John Wiley & Sons, Inc.

We show that in all dimensions >7 there are closed aspherical manifolds whose fundamental groups have nontrivial center but do not possess any topological circle actions. This disproves a conjectured converse (proposed by Conner and Raymond) to a classical theorem of Borel., 13 pages

The Goresky-MacPherson L-class of a PL pseudomanifold piecewise-linearly embedded in a PL manifold in a possibly nonlocally flat way, can be computed in terms of the Hirzebruch-Thom L-class of the manifold and twisted L-classes associated to the singularities of the embedding, as was shown by Cappell and Shaneson. These formulae are refined here by analyzing the twisted classes. We treat the case of Blanchfield local systems that extend into the singularities as well as cases where they do not extend. In the latter situation, we consider fibered embeddings of strata and 4-dimensional singular sets, using work of Banagl. Rho-invariants enter the picture.

The Milnor-Hirzebruch class of a locally complete intersection X in an algebraic manifold M measures the difference between the (Poincare dual of the) Hirzebruch class of the virtual tangent bundle of X and, respectively, the Brasselet-Schuermann-Yokura (homology) Hirzebruch class of X. In this note, we calculate the Milnor-Hirzebruch class of a globally defined algebraic hypersurface X in terms of the corresponding Hirzebruch invariants of singular strata in a Whitney stratification of X. Our approach is based on Schuermann's specialization property for the motivic Hirzebruch class transformation of Brasselet-Schuermann-Yokura. The present results also yield calculations of Todd, Chern and L-type characteristic classes of hypersurfaces., v2: title changed, exposition improved and expanded, references added

For smooth manifolds, Atiyah and Meyer studied contributions of monodromy to usual signatures. In this note we obtain Atiyah-Meyer type formulae for equivariant Hodge-theoretic genera of complex algebraic varieties. Equivariant Hirzebruch genera y(X;g) of a quasi-projective variety X acted upon by a nite group of algebraic automorphisms are dened by combining the group action with the information encoded by the Hodge ltration of the mixed Hodge structure in cohomology. While for a projective algebraic manifold y(X;g) can by computed by the Atiyah- Singer holomorphic Lefschetz theorem, we derive a Atiyah-Meyer type formula for y(X;g) in the case when X is not necessarily smooth or compact, but just bers equivariantly (in the complex topology) over a compact algebraic manifold. These results apply to computing Hodge-theoretic invariants of orbit spaces. We also obtain some results comparing equivariant Hodge-theoretic genera of the range and domain of an equivariant algebraic map in terms of its singularities.

Let X and Y be spaces and f: X -Y be a map between them. Then a large family of problems in topology (and geometry) involve the attempt to relate the invariants of X and Y via f. This attempt has been successful often for the case X and Y manifolds and f the projection of a fibre bundle, for many types of important invariants. For example, the result of ChernHirzebruch-Serre [CHS] asserts that for Y simply connected, the signature of X is the product of the signatures of Y and of the fibre f 1 (y) . In this paper we undertake the study of maps in which the fiber may vary from point to point. The spaces X and Y will be any Whitney stratified spaces with even codimension strata (e.g., algebraic varieties) and f will be a stratified map. This means essentially that for V a component of a stratum of Y (thus V is an open manifold), f 1V is a union of strata of X and the restriction of f to f71V is a locally trivial fibre bundle. For example, subanalytic maps of subanalytic sets can be stratified, with respect to Whitney

This paper announces results on the behavior of some important algebraic and topological invariants --- Euler characteristic, arithmetic genus, and their intersection homology analogues; the signature, etc. --- and their associated characteristic classes, under morphisms of projective algebraic varieties. The formulas obtained relate global invariants to singularities of general complex algebraic (or analytic) maps. These results, new even for complex manifolds, are applied to obtain a version of Grothendieck-Riemann-Roch, a calculation of Todd classes of toric varieties, and an explicit formula for the number of integral points in a polytope in Euclidean space with integral vertices., 8 pages

We show that the maximum combinatorial complexity of the space of hyperplane transversals to a family of n separated and strictly convex sets in Rd is T(n⌊d/2⌋), which generalizes results of Edelsbrunner and Sharir in the plane. As a key step in the argument, we show that the space of hyperplanes tangent to k ≤ d separated and strictly convex sets in Rd is a topological (d - k)-sphere.

global topological invariant to be studied will be the L-classes Li(X) E Hi(X; Q), X a stratified pseudomanifold with even-codimension strata. For manifolds, these characteristic classes are the Poincare duals of the Hirzebruch polynomials in the Pontrjagin classes; Goresky-MacPherson extended the definition to stratified spaces with even-codimension strata using intersection homology [GM1]. Cheeger [Ch] defined L-classes analytically using his development of L2cohomology theory of singular Riemannian spaces. See also [S], [CSW]. The importance of these classes is illustrated by the famous theorem of Browder and Novikov: The homeomorphism type of a compact simply connected manifold of dimension at least five is determined by its homotopy type and its L-classes, up to a finite number of possibilities. (For 4-manifolds, the homotopy type alone suffices [F].) By recent work of Cappell-Weinberger (see [We]), similar results hold for stratified spaces with simply connected even-codimension strata and simply connected links, with respect to isovariant homotopy type and L-classes of the space and its strata. The study of L-classes will take place in the context of a stratified pseudomanifold X7 of real dimension n, embedded, say piecewise linearly, in a manifold Mm' of dimension m = n + 2. For example X might be a hypersurface in a projective complex algebraic variety. For X a smoothly or PL locally flatly embedded submanifold, the L-classes are given by the classical formula L(X) = [X] n i*/(P(M) u (1 +X f the total Hirzebruch L-polynomial, P(M) E H*(M) the total Pontrjagin class of M, i the inclusion, and X the Poincare dual of i [X].

Chapter I : Below the Middle Dimension 1. Ambient Surgery below the Middle Dimension . . . . . . . 78 2. Review of Covering Space Theory and Related Algebra . . . .79 3. Handle Exchanges and Low Dimensions . . . . . . . . . . 86 4. Homology Computations . . . . . . . . . . . . . . . . 91 5. Completion of Proof of Lemma 1.1 . . . . . . . . . . . . 106 Appendix: Geometry of Normal Form for GI *, G, and J * , { t } 11 1

In dimension 3, the Kneser conjecture, proved in [14], implies that a P.L. manifold I$‘, homotopy equivalent to P iii Q, is itself a connected sum of manifolds homotopy equivalent to P and Q. The same situation exists in dimensions greater than 5 if P and Q are simply connected [l] or even just P simply connected [15] or in odd dimensions greater than 5 if the fundamental groups of P and Q have no elements of order 2 [Ill. In fact the same situation exists in all dimensions greater than 4 if the fundamental groups of P and Q have no elements of order 2 [4, 51. This also extends to all orientable 4X+ 3 dimensional manifolds, and to all manifolds Wzkf’ for which each element g of order 2 in n,(N’) satisfies [g] n am = 0 for k odd, I for k even, wl(W) the first Stiefel-Whitney class of CVand [g] the class in H,(IV; Z,) represented byg [5, 71.