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36
Surgery transfer
 In “Algebraic topology, Göttingen 1987.” Lecture Notes in Mathematics 1361
, 1988
"... Given a Hurewicz fibration F,E P,B with fibre an ..."
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Morse theory, graphs and string topology
"... In these lecture notes we discuss a body of work in which Morse theory is used to construct various homology and cohomology operations. In the classical setting of algebraic topology this is done by constructing a moduli space of graph flows, using homotopy theoretic methods to construct a virtual f ..."
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In these lecture notes we discuss a body of work in which Morse theory is used to construct various homology and cohomology operations. In the classical setting of algebraic topology this is done by constructing a moduli space of graph flows, using homotopy theoretic methods to construct a virtual fundamental class, and evaluating cohomology classes on this fundamental class. By using similar constructions based on “fat ” or ribbon graphs, we describe how to construct string topology operations on the loop space of a manifold, using Morse theoretic techniques. Finally, we discuss how to relate these string topology operations to the counting of J holomorphic curves in the cotangent bundle. We end with speculations about the relationship between the absolute and relative GromovWitten theory of the cotangent bundle, and the openclosed
A Manifold Calculus Approach to Link Maps and the Linking Number
, 2007
"... We study the space of link maps Link(P1,... Pk;N), the space of maps P1 · · · Pk → N such that the images of the Pi are pairwise disjoint. We apply the manifold calculus of functors developed by Goodwillie and Weiss to study the difference between it and its linear and quadratic approximations. We ..."
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We study the space of link maps Link(P1,... Pk;N), the space of maps P1 · · · Pk → N such that the images of the Pi are pairwise disjoint. We apply the manifold calculus of functors developed by Goodwillie and Weiss to study the difference between it and its linear and quadratic approximations. We identify an appropriate generalization of the linking number as the geometric object which measures the difference between the space of link maps and its linear approximation. Our analysis of the difference between link maps and its quadratic approximation connects with recent work of the author, and is used to show that the Borromean rings are linked. Key words: links, calculus of functors, linking number PACS: 1
LINKING AND COINCIDENCE INVARIANTS
, 2004
"... Abstract. Given a link map f into a manifold of the form Q = N ×R, when can it be deformed to an “unlinked ” position (in some sense, e.g. where its components map to disjoint Rlevels)? Using the language of normal bordism theory as well as the path space approach of Hatcher and Quinn we define obs ..."
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Abstract. Given a link map f into a manifold of the form Q = N ×R, when can it be deformed to an “unlinked ” position (in some sense, e.g. where its components map to disjoint Rlevels)? Using the language of normal bordism theory as well as the path space approach of Hatcher and Quinn we define obstructions ˜ωε(f), ε = + or ε = −, which often answer this question completely and which, in addition, turn out to distinguish a great number of different link homotopy classes. In certain cases they even allow a complete link homotopy classification. Our development parallels recent advances in Nielsen coincidence theory and leads also to the notion of Nielsen numbers of link maps. In the special case when N is a product of spheres sample calculations are carried out. They involve the homotopy theory of spheres and, in particular, James–Hopf– invariants.
Singularities, Double Points, Controlled Topology and Chain Duality
"... A manifold is a Poincar'e duality space without singularities. McCrory obtained a homological criterion of a global nature for deciding if a polyhedral Poincar'e duality space is a homology manifold, i.e. if the singularities are homologically inessential. A homeomorphism of manifolds is a ..."
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A manifold is a Poincar'e duality space without singularities. McCrory obtained a homological criterion of a global nature for deciding if a polyhedral Poincar'e duality space is a homology manifold, i.e. if the singularities are homologically inessential. A homeomorphism of manifolds is a degree 1 map without double points. In this paper combinatorially controlled topology and the chain complex methods of the algebraic theory of surgery are used to simplify McCrory's work, and to provide a homological criterion of a global nature for deciding if a degree 1 map of polyhedral homology manifolds has acyclic point inverses, i.e. if the double points are homologically inessential. Other applications include double points of arbitrary maps, Whitehead torsion, homology fibrations and knot theory. AMS Classification numbers Primary: 55N45, 57R67 Secondary: 55U35 Keywords: Singularities, double points, controlled topology, chain complex, duality, surgery. 1 Contents 1 Chain duality 8 2 Sim...
Higher order Nielsen numbers
"... Abstract. Suppose X, Y are manifolds, f, g: X → Y are maps. The wellknown Coincidence Problem studies the coincidence set C = {x: f(x) = g(x)}. The number m = dim X − dim Y is called the codimension of the problem. More general is the Preimage Problem. For a map f: X → Z and a submanifold Y of Z, i ..."
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Abstract. Suppose X, Y are manifolds, f, g: X → Y are maps. The wellknown Coincidence Problem studies the coincidence set C = {x: f(x) = g(x)}. The number m = dim X − dim Y is called the codimension of the problem. More general is the Preimage Problem. For a map f: X → Z and a submanifold Y of Z, it studies the preimage set C = {x: f(x) ∈ Y}, and the codimension is m = dim X + dim Y − dim Z. In case of codimension 0, the classical Nielsen number N(f, Y) is a lower estimate of the number of points in C changing under homotopies of f, and for an arbitrary codimension, of the number of components of C. We extend this theory to take into account other topological characteristics of C. The goal is to find a “lower estimate ” of the bordism group Ωp(C) of C. The answer is the Nielsen group Sp(f, Y) defined as follows. In the classical definition the Nielsen equivalence of points of C based on paths is replaced with an equivalence of singular submanifolds of C based on bordisms. We let S ′ p(f, Y) = Ωp(C) / ∼N, then the Nielsen group of order p is the part of S ′ p(f, Y) preserved under homotopies of f. The Nielsen number Np(F, Y) of order p is the rank of this group (then N(f, Y) = N0(f, Y)). These numbers are new obstructions to removability of coincidences and preimages. Some examples and computations are provided. 1.
MULTIPLE POINTS OF IMMERSIONS
, 2002
"... Abstract. Given smooth manifolds V n and M m, an integer k, and an immersion f: V � M, we have constructed an obstruction for existence of regular homotopy of f to an immersion f ′ : V � M without kfold points. This obstruction takes values in certain framed bordism group, and for (k + 1)(n + 1)≤km ..."
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Abstract. Given smooth manifolds V n and M m, an integer k, and an immersion f: V � M, we have constructed an obstruction for existence of regular homotopy of f to an immersion f ′ : V � M without kfold points. This obstruction takes values in certain framed bordism group, and for (k + 1)(n + 1)≤km turns out to be complete. 1.
Doc. Math. J. DMV 1 Singularities, Double Points, Controlled Topology and Chain Duality
, 1998
"... Abstract. A manifold is a Poincaré duality space without singularities. McCrory obtained a homological criterion of a global nature for deciding if a polyhedral Poincaré duality space is a homology manifold, i.e. if the singularities are homologically inessential. A homeomorphism of manifolds is a d ..."
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Abstract. A manifold is a Poincaré duality space without singularities. McCrory obtained a homological criterion of a global nature for deciding if a polyhedral Poincaré duality space is a homology manifold, i.e. if the singularities are homologically inessential. A homeomorphism of manifolds is a degree 1 map without double points. In this paper combinatorially controlled topology and chain complex methods are used to provide a homological criterion of a global nature for deciding if a degree 1 map of polyhedral homology manifolds has acyclic point inverses, i.e. if the double points are homologically inessential.