Sylvain Cappell - Academia.edu (original) (raw)

Papers by Sylvain Cappell

... Let L be a matrix for LV with respect to some basis. Let L'denote the transpose of L. If... more ... Let L be a matrix for LV with respect to some basis. Let L'denote the transpose of L. If£ is complex number of norm 1, Page 184. 170 SYLVAIN E. CAPPELL AND JULIUS L. SHANESON Then Kt is a Hermitian form over the complex numbers; let denote its signature. ...

Proceedings of the Royal Society of Edinburgh: Section A Mathematics

For a finite group GGG of not prime power order, Oliver showed that the obstruction for a finite ... more For a finite group GGG of not prime power order, Oliver showed that the obstruction for a finite CW-complex FFF to be the fixed point set of a contractible finite GGG -CW-complex is determined by the Euler characteristic chi(F)\chi (F)chi(F) . (He also has similar results for compact Lie group actions.) We show that the analogous problem for FFF to be the fixed point set of a finite GGG -CW-complex of some given homotopy type is still determined by the Euler characteristic. Using trace maps on K_0K_0K_0 [2, 7, 18], we also see that there are interesting roles for the fundamental group and the component structure of the fixed point set.

Proceedings, Aug 3, 2023

Smith theory says that the fixed point set of a semi-free action of a group GGG on a contractible... more Smith theory says that the fixed point set of a semi-free action of a group GGG on a contractible space is mathbbZp{\mathbb {Z}}_pmathbbZp -acyclic for any prime factor ppp of the order of GGG . Jones proved the converse of Smith theory for the case GGG is a cyclic group acting semi-freely on contractible, finite CW-complexes. We extend the theory to semi-free group actions on finite CW-complexes of given homotopy types, in various settings. In particular, the converse of Smith theory holds if and only if a certain KKK -theoretical obstruction vanishes. We also give some examples that show the geometrical effects of different types of KKK -theoretical obstructions.

arXiv (Cornell University), Sep 25, 2022

This paper studies the interaction of π1(M) for a C ∞ manifold M with Bott's original obstruction... more This paper studies the interaction of π1(M) for a C ∞ manifold M with Bott's original obstruction to integrability, and with differential geometric invariants such as Godbillon-Vey and Cheeger-Simons invariants of a foliation. We prove that the ring of higher Pontrjagin and higher Chern classes of an integrable subbundle E of the tangent bundle of a manifold vanishes above dimension 2k where k = dim(T M/E), and where the higher Pontrjagin and Chern rings are rings generated by i * y ∪ pj(T M/E) and by i * y ∪ cj(T M/E) respectively, with pj the j-th Pontrjagin class, cj the j-th Chern class, i : M → Bπ and π = π1(BG), where BG is the classifying space of the holonomy groupoid corresponding to E and y ∈ H * (Bπ), provided that the fundamental group of BG satisfies the Novikov conjecture. In addition, we show the vanishing of higher Pontrjagin and Chern rings generated by i * x ∪ pj(T M/E), and by i * x ∪ cj (T M/E) as before but with i : M → BG, BG as above and x ∈ H * (BG) provided (M, F) satisfied the foliated Novikov conjecture, where F is the foliation whose tangent bundle is E. We give examples of this obstruction and of higher Godbillon-Vey and Cheeger-Simons invariants.

Symposium on Computational Geometry, 1990

Note: Professor Pach's number: [077]; Expanded version: Common tangents and common transversa... more Note: Professor Pach's number: [077]; Expanded version: Common tangents and common transversals, Advances of Mathematics 106 (1994), 198-215 Reference DCG-CONF-2008-008 Record created on 2008-11-19, modified on 2017-05-12

Communications on Pure and Applied Mathematics, Aug 1, 1996

This paper is the first of a three-part investigation into the behavior of analytical invariants ... more 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.

arXiv (Cornell University), Mar 29, 2023

2.4. Todd and Hirzebruch classes of a simplicial toric variety 2.5. Rational equivariant cohomolo... more 2.4. Todd and Hirzebruch classes of a simplicial toric variety 2.5. Rational equivariant cohomology of a (complete simplicial) toric variety 2.6. Equivariant Euler characteristic 2.7. Equivariant Chern character and equivariant Riemann-Roch map. 3. Equivariant characteristic classes of toric varieties. 3.1. Definition. Properties 3.2. Generalized equivariant Hirzebruch-Riemann-Roch 3.3. Equivariant Hirzebruch and Todd characteristic classes of simplicial toric varieties 41 4. Localization in equivariant K-theory and applications 4.1. Localization in equivariant K-theory 4.2. Localization in equivariant homology 4.3. Equivariant Hirzebruch classes of simplicial toric varieties via localization 5. Euler-Maclaurin formulae via equivariant Hirzebruch-Riemann-Roch 5.1. Brief overview of Euler-Maclaurin formulae 5.2. Euler-Maclaurin formulae via polytope dilations 5.3. Examples of Euler-Maclaurin formulae 6. Weighted Euler-Maclaurin formulae 6.1. Abstract weighted Euler-Maclaurin formulae 6.2. Examples of weighted Euler-Maclaurin formulae 7. Euler-Maclaurin formulae via the Cappell-Shaneson algebra 7.1. Cappell-Shaneson algebra vs. completed equivariant cohomology ring 7.2. Euler-Maclaurin formulae via the Cappell-Shaneson algebra 7.3. Generalized Reciprocity for Dedekind Sums via Euler-Maclaurin formulae References

Letfbeaperiodic differentiable mapfroma sphere toitself. Awell-known conjecture ofSmith asserts t... more Letfbeaperiodic differentiable mapfroma sphere toitself. Awell-known conjecture ofSmith asserts that inmanycases (e.g., whenthefixed points areisolated) thede- rivatives offatits fixed points, regarded asJacobian matrices, arelinearly similar. Herewegivecounterexamples tothis conjecture. Theresults showthat, inmanycases, these Jacobian matrices areonly nonlinearly similar. This uses ourrecent dis- covery oforthogonal matrices which arenonlinearly similar without being linearly similar. Someresults ongeneral smooth actions offinite groups ondifferentiable manifolds arepre- sented; thetopological equivalence oftheir tangential repre- sentations atthefixed points isstudied. Awell-known conjecture ofSmith (1)states that aperiodic differentiable mapona(homology) sphere with isolated fixed points has, asderivatives atits fixed points, Jacobian matrices which arelinearly similar. Theanalogue ofthis isobvious when thefixed point set isconnected, andthis conjecture ofSmith is known inmanyimportant cases. Results ofAtiyah andBott (2) andofMilnor (3) andanextension bySanchez (4) showed the conjecture for all actions ofodd-prime power period aswell as forall maps, ofanyperiod, which give free actions outside the fixed points. Bredon (5, 6)proved that thederivatives atfixed points aresubject tosomesevere restraints. Petrie hasan- nounced that ageneralization ofthis conjecture toactions of general finite groups isfalse forcertain highly noncyclic groups. This note announces counterexamples, tothis conjecture of Smith, for each period 4q,q>1;itfurther outlines astudy and classification ofanimportant class ofcounterexamples. Inplace ofthe conjecture ofSmith, wewill seethat theJacobian matrices atthefixed points are, atleast inalarge class ofcases, nonlin- early similar matrices. Previously weannounced (7-9) the existence ofpairs ofnonlinearly similar orthogonal matrices which arenotlinearly similar, andweclassified such examples inmanycases. Thus, theconstruction ofthepresent counter- examples involves showing that someofthese pairs ofnonlin- early similar matrices arise asJacobian matrices atfixed points. Wealso study analogous questions onsmooth actions ofgeneral finite groups onmanifolds andthetangential representations attheir fixed points.

Communications on Pure and Applied Mathematics, May 1, 1999

Ú-Operator, de Rham Operator, and the Cauchy-Riemann Operator 571 6. Geometry of the Representati... more Ú-Operator, de Rham Operator, and the Cauchy-Riemann Operator 571 6. Geometry of the Representation Spaces 577 7. Walker's Correction Term and Spectral Flow 587 8. Canonical Perturbations 602 Bibliography 610 flow of a family of self-adjoint elliptic operators ´Ùµ Ä ¾´ µ Ä ¾´ µ in terms

arXiv (Cornell University), May 22, 2023

The mesh matrix Mesh(G, T 0) of a connected finite graph G = (V (G), E(G)) = (vertices, edges) of... more The mesh matrix Mesh(G, T 0) of a connected finite graph G = (V (G), E(G)) = (vertices, edges) of G of with respect to a choice of a spanning tree T 0 ⊂ G is defined and studied. It was introduced by Trent [30],[31]. Its characteristic polynomial det(X • Id − Mesh(G, T 0)) is shown to equal Σ N j=0 (−1) j ST j (G, T 0) (X − 1) N −j (⋆) where ST j (G, T 0) is the number of spanning trees of G meeting E(G − T 0) in j edges and N = |E(G − T 0)|. As a consequence, there are Tutte-type deletion-contraction formulae for computing this polynomial. Additionally, Mesh(G, T 0) − Id is of the special form Y t • Y ; so the eigenvalues of the mesh matrix Mesh(G, T 0) are all real and are furthermore be shown to be ≥ +1. It is shown that Y • Y t , called the mesh Laplacian, is a generalization of the standard graph Kirchhoff Laplacian ∆(H) = Deg − Adj of a graph H. For example, (⋆) generalizes the all minors matrix tree theorem for graphs H and gives a deletion-contraction formula for the characteristic polynomial of ∆(H). This generalization is explored in some detail. The smallest positive eigenvalue of the mesh Laplacian, a measure of flux, is estimated, thus extending the classical inequality for the Kirchoff Laplacian of graphs.

arXiv (Cornell University), Feb 21, 2007

De Gruyter eBooks, Dec 31, 1992

arXiv (Cornell University), Dec 31, 1993

This paper announces results on the behavior of some important algebraic and topological invarian... more 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.

American Journal of Mathematics, Aug 1, 1982

American Journal of Mathematics, 1975

American Journal of Mathematics, Oct 1, 1989

The real representations Pi and P2 of a finite group G are topologically similar (written Pi -t P... more The real representations Pi and P2 of a finite group G are topologically similar (written Pi -t P2) if there is a homeomorphism h: V(p1) --V(PA where V(pi) denotes the vector space of the representation pi, such that h(p1(g) * v) = P2(g) * h(v) for v e V(p1) and g E G (i.e. the representation spaces are equivariantly homeomorphic). De Rham [dR] conjectured that topological similarity implies the linear equivalence of the two representations. The first two authors showed that de Rham's conjecture is true for representations of dimension less than six [CS6], but false for representations of dimension greater than or equal to nine [CS2]. Here, we give examples of nonlinear similarity (i.e. topological similarity between linearly inequivalent representations) in dimension six, which is therefore the minimal dimension in which nonlinear similarity may occur. Since linear equivalence is detected by cyclic subgroups, a minimal counterexample to de Rham's conjecture will occur for G cyclic. Moreover, by [HP] and [MR], nonlinear similarity may only occur for groups whose order is divisible by four. Thus, we shall restrict attention to G = Z4q, the cyclic group of order 4q. We shall construct six dimensional nonlinear similarities of Z4q for every q greater than two. Moreover, we shall show in a later paper ([CSSWW2], joint with Weinberger) that the examples here represent all six-dimensional nonlinear similarities of cyclic groups. Actually, we shall give the complete topological classification of representations of the form p + 6 + c, where p is free (i.e. Z4q acts freely away from the origin in the representation space of p) and 6 and e are the nontrivial and trivial one-dimensional representations, respectively. (Representations of the form p + 6 were studied in [CS2].) Suppose that Pi + 6 + c 1t P2 + 6 + e with p1 and P2 free. Let Lp; be the orbit space of the unit

Institut des hautes études scientifiques eBooks, 1983