Mayer-Vietoris Exact Sequence Research Articles

A Twisted Version of Controlled K-theory

There are a number of (co-)homology theories on coarse spaces. Controlled operator K-theory is by far the most popular one of them. Our approach is geometric. We study when does the Roe-algebra of a space restrict to a subspace. Then we show the Roe-algebra is a cosheaf on the coarse topology. A result is a Mayer–Vietoris exact sequence in the presence of a coarse cover. We compute examples as an application.

Symplectic homology and the Eilenberg–Steenrod axioms

We give a definition of symplectic homology for pairs of filled Liouville cobordisms, and show that it satisfies analogues of the Eilenberg-Steenrod axioms except for the dimension axiom. The resulting long exact sequence of a pair generalizes various earlier long exact sequences such as the handle attaching sequence, the Legendrian duality sequence, and the exact sequence relating symplectic homology and Rabinowitz Floer homology. New consequences of this framework include a Mayer-Vietoris exact sequence for symplectic homology, invariance of Rabinowitz Floer homology under subcritical handle attachment, and a new product on Rabinowitz Floer homology unifying the pair-of-pants product on symplectic homology with a secondary coproduct on positive symplectic homology. In the appendix, joint with Peter Albers, we discuss obstructions to the existence of certain Liouville cobordisms.

Scattering matrix and analytic torsion

We consider a compact manifold with a piece isometric to a (finite-length) cylinder. By making the length of the cylinder tend to infinity, we obtain an asymptotic gluing formula for the zeta determinant of the Hodge Laplacian and an asymptotic expansion of the L2 torsion of the corresponding Mayer–Vietoris exact sequence. As an application, we give a purely analytic proof of the gluing formula for analytic torsion.

Controlled Topological Phases and Bulk-edge Correspondence

In this paper, we introduce a variation of the notion of topological phase reflecting metric structure of the position space. This framework contains not only periodic and non-periodic systems with symmetries in Kitaev's periodic table but also topological crystalline insulators. We also define the bulk and edge indices as invariants taking values in the twisted equivariant $\mathrm{K}$-groups of Roe algebras as generalizations of existing invariants such as the Hall conductance or the Kane--Mele $\mathbb{Z}_2$-invariant. As a consequence, we obtain a new mathematical proof of the bulk-edge correspondence by using the coarse Mayer-Vietoris exact sequence.

Discrete homology theory for metric spaces

In this paper we define and study a notion of discrete homology theory for metric spaces. Instead of working with simplicial homology, our chain complexes are given by Lipschitz maps from an $n$-dimensional cube to a fixed metric space. We prove that the resulting homology theory verifies a discrete analogue of the Eilenberg-Steenrod axioms, and prove a discrete analogue of the Mayer-Vietoris exact sequence. Moreover, this discrete homology theory is related to the discrete homotopy theory of a metric space through a discrete analogue of the Hurewicz theorem. We study the class of groups that can arise as discrete homology groups and, in this setting, we prove that the fundamental group of a smooth, connected, metrizable, compact manifold is isomorphic to the discrete fundamental group of a `fine enough' rectangulation of the manifold. Finally, we show that this discrete homology theory can be coarsened, leading to a new non-trivial coarse invariant of a metric space.

La suite exacte de Mayer–Vietoris en cohomologie de Čech

This article is about the foundation of the Mayer–Vietoris exact sequence in Čech cohomology, with respect to the augmented Čech complex, often called “the stable Koszul complex”. Our treatment is elementary and uses neither the local cohomology of Grothendieck nor the widespread noetherian local cohomology.We use explicit objects giving us an elementary algebraic treatment in a general nonnoetherian context, i.e. two finite sequences a̲, b̲ of a commutative ring and some Čech complexes built from these sequences. More precisely, our strategy consists in producing a short exact sequence of “type Čech“ complexes having the expected cohomologies. Then arises a simplicial complex on the set (a̲,b̲,ab̲) with a relatively surprising combinatory.

Algebraic K-theory and cyclic homology

The following are personal reminiscences of my research years in algebraic K-theory and cyclic homology during which Dan Quillen was everyday present in my professional life.In the late sixties (of the twentieth century) the groups K0;K1;K2 were known and well-studied. The group K0 had been introduced by Alexander Grothendieck, then came K1 by Hyman Bass [2] (as a variation of the Whitehead group), permitting one to generalize the notion of determinant, and finally K2 by John Milnor [9] and Michel Kervaire. The big problem was: how about Kn? Having in mind topological K-theory and all the other generalized (co)homological theories, one was expecting higher K-groups which satisfy similar axioms, in particular the Mayer-Vietoris exact sequence. The discovery by Richard Swan of the existence of an obstruction for this property to hold shed some embarrassment. What kind of properties should we ask of Kn? There were various attempts, for instance by Max Karoubi and Orlando Villamayor [4]. And suddenly Dan Quillen came with a candidate sharing a lot of nice properties. He had even two different constructions of the same object: the “+” construction and the “Q” construction [14, 15]. Not only did he propose a candidate but he already got a computation: the higher K-theory of finite fields. This was a fantastic step forward and Hyman Bass organized a two week conference at the Battelle Institute in Seattle during the summer of 1972, which was attended by Bass, Borel, Husemoller, Karoubi, Priddy, Quillen, Segal, Stasheff, Tate, Waldhausen, Wall and sixty other mathematicians. The Proceedings appeared as Springer Lecture Notes 341, 342 and 343. I met Quillen for the first time on this occasion.

Sur la K-théorie du foncteur norme

The K-theory of a functor may be viewed as a relative version of the K-theory of a ring. After its description, we prove a Mayer–Vietoris exact sequence in this framework. In the case of a Galois extension of a number field F / L with rings of integers A, B respectively, this K-theory of the “norm functor” is an extension of a subgroup of the ideal class group Cl ( A ) of F by the Tate cohomology group H ˆ 0 ( G , A ∗ ) . The Mayer–Vietoris exact sequence enables us to describe in a quite explicit way a quotient of the subgroup Cl N ( A ) : = ker N : Cl ( A ) → Cl ( B ) of the ideal class group Cl ( A ) , where N is the norm. We also prove a short exact sequence 1 → B ∗ / B ∗ ∩ N ( F ∗ ) → H ˆ 0 ( G , F ∗ ) ∩ H ˆ 0 ( G , U F ) → Cl N ( A ) / I G Cl ( A ) → 1 where U F is the group of semi-local units of F. Finally, we conclude this paper by applications of our methods to Number Theory.

Suite exacte de Mayer-Vietoris d'une extension triangulaire

In this article, we complete the results published in [2] using new proofs: we obtain a more general long exact sequence for the extension groups $\operatorname {Ext}_T^*$ of the triangular algebra $T=\left[\begin{smallmatrix} A & M \\ 0 & B\end{smallmatrix}\right]$. One may find such a long exact sequence in the works of Palmér and Roos [22], [23], but only for very special instances of $T$-modules. This extra effort allows us to show that the well-known Hochschild cohomological exact sequence of Happel ([16] and, more generally,[8] and [20]) is just a particular instance of that of Palmér-Roos, even if it seems we really need an assumption on $M$ (the same trick also appears for the homological version using the ${\operatorname {Tor}}^T_*$ functor). With this method, we obtain a sensible improvement for the isomorphism of [3], which computes the Hochschild cohomology of triangular algebras in several cases. At the end of the article, we relate the Palmér-Roos exact sequence with the classical (i.e. topological) Mayer-Vietoris exact sequence.

GLUING TECHNIQUES IN TRIANGULAR GEOMETRY

We discuss gluing of objects and gluing of morphisms in triangulated categories. We illustrate the results by producing, among other things, a Mayer-Vietoris exact sequence involving Picard groups.

A degenerate Newton’s map in two complex variables: Linking with currents

Little is known about the global structure of the basins of attraction of Newton’s method in two or more complex variables. We make the first steps by focusing on the specific Newton mapping to solve for the common roots of P(x,y) = x(1 x) and Q(x,y) = y 2 + Bxy y. There are invariant circles S0 and S1 within the lines x = 0 and x = 1 which are superattracting in the x-direction and hyperbolically repelling within the vertical line. We show that S0 and S1 have local super-stable manifolds, which when pulled back under iterates of N form global super-stable spaces W0 and W1. By blowing-up the points of indeterminacy p and q of N and all of their inverse images under N we prove that W0 and W1 are real-analytic varieties. We define linking between closed 1-cycles in Wi (i = 0,1) and an appropriate closed 2 current providing a homomorphism lk : H1(Wi, Z) ! Q. If Wi intersects the critical value locus of N, this homomorphism has dense image, proving that H1(Wi, Z) is infinitely generated. Using the Mayer-Vietoris exact sequence and an algebraic trick, we show that the same is true for the closures of the basins of the roots W(ri).

Cohomological lower bounds for isoperimetric functions on groups

If the finitely presented group G splits over the finitely presented sub-group C, then classes are constructed in H 2 (∞) (G) which reflect the splitting and which serve as lower bounds for isoperimetric functions for G. It is proved that H 2 (∞) (G)=0 for all word hyperbolic groups G. A converse is obtained for the combination theorem for hyperbolic groups of Bestvina–Feighn. The Mayer–Vietoris exact sequence for l ∞ -cohomology associated to a splitting of a group is established. Metabolic groups are introduced as finitely presented groups G such that H 2 (∞) (G, A)=0 for all normed abelian coefficient groups A and such groups G are shown to be characterized by possessing “thin” combings.

A spectral sequence for the intersection of subspace pairs

A general-homology spectral sequence that generalizes the Mayer-Vietoris exact sequence is established between the intersection of a family of subspace pairs and the system of partial unions of the family. The basis of the is a topological analogue of the bar construction of homological algebra. We shall show here that a finite family = {(Xi, A )Ii E I} of subspace pairs' in a space X have, for each general homology theory h*, a spectral sequence (a) E n_ hj U Xi, U Ai) => hjn(n Xi n A) Ns=n s s I I (sc: I), Ns being (number of members in s)-1. This is just the spectral sequence of a cover with the roles of union and intersection interchanged. Its connection with the Mayer-Vietoris sequence will be examined below, and we shall derive from it the spectral sequence of the homology sheaf of X. Construction of (a). Start with any finite set U (= the universe) that contains I as a subset, and define (using T to denote the based unit interval, while Ai Y= YJ1{2 E YJj* E nJ} for based spaces Y and finite sets J) Vs =A aTAAT (s c U), s U-s K = UV{a} = aVz aEU C= {*} U U V{a}= ATAaAT, (b) aeU-I I U-I M=X x CUU UAi x Vs, sCI s Ln=Mu U Ux,xVs (neZ) scl?;Ns!n s L = = X x C U U U X, X Vs. sCI s Received by the editors February 16, 1973. AMS (MOS) subject classifications (1970). Primary 55H05, 55H25; Secondary 55C05.