Surgery Exact Sequence Research Articles

Chain duality for categories over complexes

We show that the additive category of chain complexes parametrized by a finite simplicial complex K forms a category with chain duality. This fact, never fully proven in the original reference (Ranicki, 1992), is fundamental for Ranicki's algebraic formulation of the surgery exact sequence of Sullivan and Wall, and his interpretation of the surgery obstruction map as the passage from local Poincaré duality to global Poincaré duality. Our paper also gives a new, conceptual, and geometric treatment of chain duality on K -based chain complexes.

Codimension 2 transfer of higher index invariants

This paper is devoted to the study of the higher index theory of codimension 2 submanifolds originated by Gromov–Lawson and Hanke–Pape–Schick. The first main result is to construct the ‘codimension 2 transfer’ map from the Higson–Roe analytic surgery exact sequence of a manifold M to that of its codimension 2 submanifold N under some assumptions on homotopy groups. This map sends the primary and secondary higher index invariants of M to those of N. The second is to establish that the codimension 2 transfer map is adjoint to the co-transfer map in cyclic cohomology, defined by the cup product with a group cocycle. This relates the Connes–Moscovici higher index pairing and Lott’s higher $$\rho $$ -number of M with those of N.

Stratified surgery and K-theory invariants of the signature operator

In work of Higson-Roe the fundamental role of the signature as a homotopy and bordism invariant for oriented manifolds is made manifest in how it and related secondary invariants define a natural transformation between the (Browder-Novikov-Sullivan-Wall) surgery exact sequence and a long exact sequence of C*-algebra K-theory groups. In recent years the (higher) signature invariants have been extended from closed oriented manifolds to a class of stratified spaces known as L-spaces or Cheeger spaces. In this paper we show that secondary invariants, such as the rho-class, also extend from closed manifolds to Cheeger spaces. We revisit a surgery exact sequence for stratified spaces originally introduced by Browder-Quinn and obtain a natural transformation analogous to that of Higson-Roe. We also discuss geometric applications.

The adiabatic groupoid and the Higson–Roe exact sequence

Let \widetilde{X} be a smooth Riemannian manifold equipped with a proper, free, isometric, and cocompact action of a discrete group \Gamma . In this paper, we prove that the analytic surgery exact sequence of Higson–Roe for \widetilde{X} is isomorphic to the exact sequence associated to the adiabatic deformation of the Lie groupoid \widetilde{X}\times_\Gamma\widetilde{X} . We then generalize this result to the context of smoothly stratified manifolds. Finally, we show, by means of the aforementioned isomorphism, that the \varrho -classes associated to a metric with a positive scalar curvature defined by Piazza and Schick (2014) correspond to the \varrho -classes defined by Zenobi (2019).

The Higson–Roe sequence for étale groupoids. II. The universal sequence for equivariant families

This is the second part of our series about the Higson–Roe sequence for étale groupoids. We devote this part to the proof of the universal K-theory surgery exact sequence which extends the seminal results of N. Higson and J. Roe to the case of transformation groupoids. In the process, we prove the expected functoriality properties as well as the Paschke–Higson duality theorem.

Mapping the surgery exact sequence for topological manifolds to analysis

In this paper we prove the existence of a natural mapping from the surgery exact sequence for topological manifolds to the analytic surgery exact sequence of Higson and Roe. This generalizes the fundamental result of Higson and Roe, but in the treatment given by Piazza and Schick, from smooth manifolds to topological manifolds. Crucial to our treatment is the Lipschitz signature operator of Teleman. We also give a generalization to the equivariant setting of the product defined by Siegel in his Ph.D. thesis. Geometric applications are given to stability results for rho classes. We also obtain a proof of the APS delocalized index theorem on odd dimensional manifolds, both for the spin Dirac operator and the signature operator, thus extending to odd dimensions the results of Piazza and Schick. Consequently, we are able to discuss the mapping of the surgery sequence in all dimensions.

Mapping Surgery To Analysis III: Exact Sequences

Using the constructions of the preceding two papers, we construct a natural transformation (after inverting 2) from the Browder-Novikov-Sullivan-Wall surgery exact sequence of a compact manifold to a certain exact sequence of C ∗ -algebra K-theory groups. Mathematics Subject Classifications (1991): 19J25, 19K99.

Realizing the analytic surgery group of Higson and Roe geometrically part III: higher invariants

We construct an isomorphism between the geometric model and Higson-Roe's analytic surgery group, reconciling the constructions in the previous papers in the series on Realizing the analytic surgery group of Higson and Roe geometrically with their analytic counterparts. Following work of Lott and Wahl, we construct a Chern character on the geometric model for the surgery group; it is a Chern character, from which Lott's higher delocalized $\rho$-invariants can be retrieved. Following work of Piazza and Schick, we construct a geometric map from Stolz' positive scalar curvature sequence to the geometric model of Higson-Roe's analytic surgery exact sequence.

Realizing the analytic surgery group of Higson and Roe geometrically part II: relative $$\eta $$ η -invariants

© 2016, Springer-Verlag Berlin Heidelberg. We construct an isomorphism between the geometric model and Higson-Roe’s analytic surgery group, reconciling the constructions in the previous papers in the series on “Realizing the analytic surgery group of Higson and Roe geometrically” with their analytic counterparts. Following work of Lott and Wahl, we construct a Chern character on the geometric model for the surgery group; it is a “delocalized Chern character”, from which Lott’s higher delocalized ρ-invariants can be retrieved. Following work of Piazza and Schick, we construct a geometric map from Stolz’ positive scalar curvature sequence to the geometric model of Higson-Roe’s analytic surgery exact sequence.

The surgery exact sequence, K-theory and the signature operator

The main result of this paper is a new and direct proof of the natural transformation from the surgery exact sequence in topology to the analytic K-theory sequence of Higson and Roe. Our approach makes crucial use of analytic properties and new index theorems for the signature operator on Galois coverings with boundary. These are of independent interest and form the second main theme of the paper. The main technical novelty is the use of large scale index theory for Dirac type operators that are perturbed by lower order operators.

From $$({\mathbb {Z}},X)$$ ( Z , X ) -modules to homotopy cosheaves

We construct a functor from the category of $(\mathbb{Z},X)$-modules of Ranicki (cf. \cite{Ra92}) to the category of homotopy cosheaves of chain complexes of Ranicki-Weiss (cf. \cite{RaWei10}) inducing an equivalence on $L$-theory. The $L$-theory of $(\mathbb{Z},X)$-modules is central in the algebraic formulation of the surgery exact sequence and in the construction of the total surgery obstruction by Ranicki, as described in \cite{Ra79}. The symmetric $L$-theory of homotopy cosheaf complexes is used by Ranicki-Weiss in \cite{RaWei10}, to reprove the topological invariance of rational Pontryagin classes. The work presented here may be considered as an addendum to the latter article and suggests some translation of ideas of Ranicki into the language of homotopy chain complexes of cosheaves.

Rho-classes, index theory and Stolz’ positive scalar curvature sequence

In this paper, we study the space of metrics of positive scalar curvature using methods from coarse geometry. Given a closed spin manifold M with fundamental group G, Stephan Stolz introduced the positive scalar curvature exact sequence, in analogy to the surgery exact sequence in topology. It calculates a structure group of metrics of positive scalar curvature on M (the object we want to understand) in terms of spin-bordism of BG and a somewhat mysterious group R(G). Higson and Roe introduced a K-theory exact sequence in coarse geometry which contains the Baum-Connes assembly map, with one crucial term K(D*G) canonically associated to G. The K-theory groups in question are the home of interesting index invariants and secondary invariants, in particular the rho-class in K_*(D*G) of a metric of positive scalar curvature on a spin manifold. One of our main results is the construction of a map from the Stolz exact sequence to the Higson-Roe exact sequence (commuting with all arrows), using coarse index theory throughout. Our main tool are two index theorems, which we believe to be of independent interest. The first is an index theorem of Atiyah-Patodi-Singer type. Here, assume that Y is a compact spin manifold with boundary, with a Riemannian metric g which is of positive scalar curvature when restricted to the boundary (and with fundamental group G). Because the Dirac operator on the boundary is invertible, one constructs a delocalized APS-index in K_* (D*G). We then show that this class equals the rho-class of the boundary. The second theorem equates a partitioned manifold rho-class of a positive scalar curvature metric to the rho-class of the partitioning hypersurface.

The smooth structure set of S p × S q

We calculate $${\mathcal{S}^{{\it Diff}}(S^p \times S^q)}$$ , the smooth structure set of S p × S q , for p, q ≥ 2 and p + q ≥ 5. As a consequence we show that in general $${\mathcal{S}^{Diff}(S^{4j-1}\times S^{4k})}$$ cannot admit a group structure such that the smooth surgery exact sequence is a long exact sequence of groups. We also show that the image of the forgetful map $${\mathcal{S}^{Diff}(S^{4j}\times S^{4k}) \rightarrow \mathcal{S}^{Top}(S^{4j}\times S^{4k})}$$ is not in general a subgroup of the topological structure set.

Epsilon–delta surgery over $${\mathbb{Z}}$$

The purpose of this paper is to prove a controlled surgery exact sequence, including a stability theorem, as used in the construction of exotic homology manifolds. The approach is to show that this result is a formal consequence of the Chapman–Ferry Alpha-approximation theorem.

Structure sets of triples of manifolds

The structure set of a given manifold fits into a surgery exact sequence, which is the main tool for classification of manifolds. In the present paper, we describe relations between various structure sets and groups of obstructions which naturally arise for triples of manifolds. The main results are given by commutative braids and diagrams of exact sequences.

Mini-Workshop: The Hauptvermutung for High-Dimensional Manifolds

The Mini-Workshop The Hauptvermutung for High-Dimensional Manifolds , organised by Erik Pedersen (Binghamton) and Andrew Ranicki (Edinburgh) was held August 13th–18th, 2006. The meeting was attended by 17 participants, ranging from graduate students to seasoned veterans. The manifold Hauptvermutung is the conjecture that topological manifolds have a unique combinatorial structure. This conjecture was disproved in 1969 by Kirby and Siebenmann, who used a mixture of geometric and algebraic methods to classify the combinatorial structures on manifolds of dimension >4 . However, there is some dissatisfaction in the community with the state of the literature on this topic. This has been voiced most forcefully by Novikov, who has written “In particular, the final Kirby–Siebenmann classification of topological multidimensional manifolds therefore is not proved yet in the literature.” () At this conference we discussed a number of questions concerning the Hauptvermutung and the structure theory of high-dimensional topological manifolds. These are our conclusions: We found nothing fundamentally wrong with the original work of Kirby and Siebenmann [4], which is solidly grounded in the literature. Their determination of \textit{TOP}/\textit{PL} depends on Kirby's paper on the Annulus Conjecture and his ‘torus trick’. It was noted that Kirby's paper is based on the well-documented work on \textit{PL} classification of homotopy tori (Hsiang and Shaneson, Wall) and Sullivan's identification of the \textit{PL} normal invariants with [-,G/\textit{PL}] , but does not depend on any other work of Sullivan, documented or undocumented. This classification can be reduced to the Farrell Fibering Theorem [1], the calculation of \pi_i(G/\textit{PL}) (Kervaire and Milnor [3]), and Wall's non-simply connected surgery theory [7]. There are modern proofs determining the homotopy type of \textit{TOP}/\textit{PL} using either the bounded surgery of Ferry and Pedersen [2] or a modification of the definition of the structure set. Sullivan's determination of the homotopy type of G/\textit{PL} , which is well documented (for instance, in Madsen and Milgram [5]) is used to determine the homotopy type of G/\textit{TOP} and is fundamental to understanding the classification of general topological manifolds. The 4-fold periodicity of the topological surgery sequence established by Siebenmann [4, p.283] contains a minor error having to do with base points. This is an easily corrected error, and the 4-fold periodicity is true whenever the manifold has a boundary. The equivalence of the algebraic and topological surgery exact sequence as established by Ranicki [6] was confirmed. Sullivan's characteristic variety theorem, however it is understood, is not essential for the Kirby–Siebenmann triangulation of manifolds. The following papers have been commissioned:

On the splitting problem for manifold pairs with boundaries

The problem of splitting a homotopy equivalence along a submanifold is closely related to the surgery exact sequence and to the problem of surgery of manifold pairs. In classical surgery theory there exist two approaches to surgery in the category of manifolds with boundaries. In the $ rel \partial$ case the surgery on a manifold pair is considered with the given fixed manifold structure on the boundary. In the relative case the surgery on the manifold with boundary is considered without fixing maps on the boundary. Consider a normal map to a manifold pair $(Y, \partial Y)\subset (X, \partial X)$ with boundary which is a simple homotopy equivalence on the boundary $\partial X$. This map defines a mixed structure on the manifold with the boundary in the sense of Wall. We introduce and study groups of obstructions to splitting of such mixed structures along submanifold with boundary $(Y, \partial Y)$. We describe relations of these groups to classical surgery and splitting obstruction groups. We also consider several geometric examples.

Relative groups in surgery theory

In this paper we consider various types of relative groups which naturally arise in surgery theory, and describe algebraic properties of them. Then we apply the obtained results to investigate the splitting obstruction groups $LS_*$ and the surgery obstruction groups $LP_*$ for a manifold pair. Finally, we introduce the lower $LS_*$- and $LP_*$-groups, and describe connections between them and the corresponding lower $L_*$-groups and surgery exact sequence.

Homology manifold bordism

The Bryant-Ferry-Mio-Weinberger surgery exact sequence for compact A N R ANR homology manifolds of dimension ≥ 6 \geq 6 is used to obtain transversality, splitting and bordism results for homology manifolds, generalizing previous work of Johnston. First, we establish homology manifold transversality for submanifolds of dimension ≥ 7 \geq 7 : if f : M → P f:M \to P is a map from an m m -dimensional homology manifold M M to a space P P , and Q ⊂ P Q \subset P is a subspace with a topological q q -block bundle neighborhood, and m − q ≥ 7 m-q \geq 7 , then f f is homology manifold s s -cobordant to a map which is transverse to Q Q , with f − 1 ( Q ) ⊂ M f^{-1}(Q) \subset M an ( m − q ) (m-q) -dimensional homology submanifold. Second, we obtain a codimension q q splitting obstruction s Q ( f ) ∈ L S m − q ( Φ ) s_Q(f) \in LS_{m-q}(\Phi ) in the Wall L S LS -group for a simple homotopy equivalence f : M → P f:M \to P from an m m -dimensional homology manifold M M to an m m -dimensional Poincaré space P P with a codimension q q Poincaré subspace Q ⊂ P Q \subset P with a topological normal bundle, such that s Q ( f ) = 0 s_Q(f)=0 if (and for m − q ≥ 7 m-q \geq 7 only if) f f splits at Q Q up to homology manifold s s -cobordism. Third, we obtain the multiplicative structure of the homology manifold bordism groups Ω ∗ H ≅ Ω ∗ T O P [ L 0 ( Z ) ] \Omega ^H_*\cong \Omega ^{TOP}_*[L_0(\mathbb Z)] .

A surgery sequence in dimension four; the relations with knot concordance

We present a systematic treatment of the classification problem for compact smooth 4-manifolds M. It is modeled on the surgery exact sequence, the central theorem in the classification of n-manifolds n>5. The price for the extension to dimension=4 is a hole in M where a (homotopy) 1-skeleton should be. There is no homotopy theoretic or surgical obstruction to complet- ing M with a wedge of circles so that the completion has the topology of a compact smooth manifold. This point-set problem is all that stands between 4- manifolds and the tranquility that prevails in higher dimensions. When M is simply connected only a point is missing from the model. The applications of this are discussed in (F) and (FQ 1). The general theory has application to knot and link theory. In particular, knots with Alexander polynomial= 1 are characterized geometrically as knots admitting a certain