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20
EpsilonDelta surgery over Z
, 2003
"... This manuscript fills in the details of the lecture I gave on “squeezing structures in Trieste in June, 2001. The goal is to develop a controlled surgery theory of the sort discussed/used in [3]. This material will appear as part of the writeup of my CBMS lectures. ..."
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This manuscript fills in the details of the lecture I gave on “squeezing structures in Trieste in June, 2001. The goal is to develop a controlled surgery theory of the sort discussed/used in [3]. This material will appear as part of the writeup of my CBMS lectures.
THE BINGBORSUK AND THE BUSEMANN CONJECTURES
, 2009
"... We present two classical conjectures concerning the characterization of manifolds: the Bing Borsuk Conjecture asserts that every ndimensional homogeneous ANR is a topological nmanifold, whereas the Busemann Conjecture asserts that every ndimensional Gspace is a topological nmanifold. The key ob ..."
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We present two classical conjectures concerning the characterization of manifolds: the Bing Borsuk Conjecture asserts that every ndimensional homogeneous ANR is a topological nmanifold, whereas the Busemann Conjecture asserts that every ndimensional Gspace is a topological nmanifold. The key object in both cases are socalled generalized manifolds, i.e. ENR homology manifolds. We look at the history, from the early beginnings to the present day. We also list several open problems and related conjectures.
Controlled L–theory
"... We develop an epsiloncontrolled algebraic L–theory, extending our earlier work on epsiloncontrolled algebraic K –theory. The controlled L–theory is very close to being a generalized homology theory; we study analogues of the homology exact sequence of a pair, excision properties, and the Mayer–Vie ..."
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We develop an epsiloncontrolled algebraic L–theory, extending our earlier work on epsiloncontrolled algebraic K –theory. The controlled L–theory is very close to being a generalized homology theory; we study analogues of the homology exact sequence of a pair, excision properties, and the Mayer–Vietoris exact sequence. As an application we give a controlled L–theory proof of the classic theorem of Novikov on the topological invariance of the rational Pontrjagin classes. 57R67; 18F25 1
Hyperbolic knots and 4–dimensional surgery
, 2006
"... In [5], Hegenbarth and Repovs ̌ used controlled surgery exact sequence of [6] to show that the surgery obstruction theory works for certain 4manifolds without assuming that the fundamental groups are good. Among their examples are 4manifolds whose fundamental groups are knot groups. Let K be a kno ..."
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In [5], Hegenbarth and Repovs ̌ used controlled surgery exact sequence of [6] to show that the surgery obstruction theory works for certain 4manifolds without assuming that the fundamental groups are good. Among their examples are 4manifolds whose fundamental groups are knot groups. Let K be a knot in S3, and
APPLICATIONS OF CONTROLLED SURGERY IN DIMENSION 4: EXAMPLES
, 2006
"... Abstract. The validity of Freedman’s disk theorem is known to depend only on the fundamental group. It was conjectured that it fails for nonabelian free fundamental groups. If this were true then surgery theory would work in dimension four. Recently, Krushkal and Lee proved a surprising result that ..."
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Abstract. The validity of Freedman’s disk theorem is known to depend only on the fundamental group. It was conjectured that it fails for nonabelian free fundamental groups. If this were true then surgery theory would work in dimension four. Recently, Krushkal and Lee proved a surprising result that surgery theory works for a large special class of 4manifolds with free nonabelian fundamental groups. The goal of this paper is to show that this also holds for other fundamental groups which are not known to be good, and that it is best understood using controlled surgery theory of Pedersen–Quinn–Ranicki. We consider some examples of 4manifolds which have the fundamental group either of a closed aspherical surface or of a 3dimensional knot space. A more general theorem is stated in the appendix.
The Bryant–Ferry–Mio–Weinberger construction of generalized manifolds, Exotic Homology Manifolds, Oberwolfach 2003, Frank Quinn and Andrew Ranicki, Eds., Geometry and Topology Monographs 9
, 2006
"... spaces. The basic ingredient is the "ı–surgery sequence recently proved by Pedersen, Quinn and Ranicki. Since one has to apply it not only in cases when the target is a manifold, but a controlled Poincaré complex, we explain this issue very roughly. Specifically, it is applied in the inductive ..."
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spaces. The basic ingredient is the "ı–surgery sequence recently proved by Pedersen, Quinn and Ranicki. Since one has to apply it not only in cases when the target is a manifold, but a controlled Poincaré complex, we explain this issue very roughly. Specifically, it is applied in the inductive step to construct the desired controlled homotopy equivalence piC1W XiC1!Xi. Our main theorem requires a sufficiently controlled Poincaré structure on Xi (over Xi1). Our construction shows that this can be achieved. In fact, the Poincaré structure of Xi depends upon a homotopy equivalence used to glue two manifold pieces together (the rest is surgery theory leaving unaltered the Poincaré structure). It follows from the "ı–surgery sequence (more precisely from the Wall realization part) that this homotopy equivalence is sufficiently well controlled. In the final section we give additional explanation why the limit space of the Xi ’s has no resolution.
Desingularizing homology manifolds
 Geom. Topol
"... We prove that if X n, n 6, is a compact ANR homology n–manifold, we can blow up the singularities of X to obtain an ANR homology n–manifold with the disjoint disks property. More precisely, we show that there is an ANR homology n–manifold Y with the disjoint disks property and a celllike map f W Y! ..."
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We prove that if X n, n 6, is a compact ANR homology n–manifold, we can blow up the singularities of X to obtain an ANR homology n–manifold with the disjoint disks property. More precisely, we show that there is an ANR homology n–manifold Y with the disjoint disks property and a celllike map f W Y! X. 57N15, 57P99 1
The Structure Set of an Arbitrary Space, the Algebraic Surgery Exact Sequence and the Total Surgery Obstruction
, 2001
"... ..."
SOME RECENT APPROACHES IN 4–DIMENSIONAL SURGERY THEORY
, 2005
"... It is wellknown that an ndimensional Poincaré complex X n, n ≥ 5, has the homotopy type of a compact topological n–manifold if the total surgery obstruction s(Xn) vanishes. The present paper discusses recent attempts to prove analogous result in dimension 4. We begin by reviewing the necessary alg ..."
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It is wellknown that an ndimensional Poincaré complex X n, n ≥ 5, has the homotopy type of a compact topological n–manifold if the total surgery obstruction s(Xn) vanishes. The present paper discusses recent attempts to prove analogous result in dimension 4. We begin by reviewing the necessary algebraic and controlled surgery theory. Next, we discuss the key idea of Quinn’s approach. Finally, we present some cases of special fundamental groups, due to the authors and to Yamasaki.
Controlled Surgery with Good Local Fundamental Groups
"... The aim of this talk is to discuss a possibility to extend the following controlled surgery exact sequence: Theorem [PQR] (simplified version) Suppose B is a finite dimensional compact metric ANR, and a dimension n ≥ 4 is given. Then there exists a number 0> 0 which depends on B and n so that for ..."
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The aim of this talk is to discuss a possibility to extend the following controlled surgery exact sequence: Theorem [PQR] (simplified version) Suppose B is a finite dimensional compact metric ANR, and a dimension n ≥ 4 is given. Then there exists a number 0> 0 which depends on B and n so that for any 0> > 0 there is δ> 0 with the following property: If p: X → B is UV 1 and X is a closed topological nmanifold then there is a controlled surgery exact sequence Hn+1(B,L) → S,δ(X, p) → [X,G/TOP] → Hn(B,L). For n ≥ 5, it seems that the above should hold true for reasonably good control maps (e.g. stratified systems of fibrations) p: X → B, if one replaces the homology groupsHi(B,L) with the controlled Lgroups Lci (B, p). This may be obvious for experts, but not for me. I will really appreciate it if someone can help me writing down the detailed proof of the controlled surgery exact sequence in this generality. The reason we have homology when p is UV 1 is that the controlled Whitehead group Whc(B, p) vanihes for such p, and this in turn comes from the fact that the ordinary