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16
Embedding and knotting of manifolds in Euclidean spaces
 London Math. Soc. Lect. Notes
"... Abstract. A clear understanding of topology of higherdimensional objects is important in many branches of both pure and applied mathematics. In this survey we attempt to present some results of higherdimensional topology in a way which makes clear the visual and algebraic constructions appear natu ..."
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Cited by 12 (7 self)
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Abstract. A clear understanding of topology of higherdimensional objects is important in many branches of both pure and applied mathematics. In this survey we attempt to present some results of higherdimensional topology in a way which makes clear the visual and algebraic constructions appear naturally in the study of geometric problems. Before giving a general construction, we illustrate the main ideas in simple but important particular cases, in which the essence is not veiled by technicalities. More specifically, we present several classical and modern results on the embedding and knotting of manifolds in Euclidean space. We state many concrete results (in particular, recent explicit classification of knotted tori). Their statements (but not proofs!) are simple and accessible to nonspecialists. We outline a general approach to embeddings via the classical van KampenShapiroWuHaefligerWeber ’deleted product ’ obstruction. This approach reduces the isotopy classification of embeddings to the homotopy classification of equivariant maps, and so implies the above concrete results. We describe the revival of interest in this beautiful branch of topology, by presenting new results in this area (of Freedman, Krushkal, Teichner, Segal, Spie˙z and the author): a generalization the HaefligerWeber embedding theorem below the metastable dimension range and examples showing that other analogues of this theorem are false outside the metastable dimension range. 1.
Einstein metrics and the number of smooth structures on a fourmanifold
, 2003
"... We prove that for every natural number k there are simply connected topological four–manifolds which have at least k distinct smooth structures supporting Einstein metrics, and also have infinitely many distinct smooth structures not supporting Einstein metrics. Moreover, all these smooth structur ..."
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Cited by 5 (2 self)
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We prove that for every natural number k there are simply connected topological four–manifolds which have at least k distinct smooth structures supporting Einstein metrics, and also have infinitely many distinct smooth structures not supporting Einstein metrics. Moreover, all these smooth structures become diffeomorphic to each other after connected sum with only one copy of the complex projective plane. We prove that manifolds with these properties cover a large geographical area.
Dissolving fourmanifolds and positive scalar curvature
 Math. Z
"... ABSTRACT. We prove that many simply connected symplectic fourmanifolds dissolve after connected sum with only one copy of S 2 × S 2. For any finite group G that acts freely on the threesphere we construct closed smooth fourmanifolds with fundamental group G which do not admit metrics of positive s ..."
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Cited by 3 (2 self)
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ABSTRACT. We prove that many simply connected symplectic fourmanifolds dissolve after connected sum with only one copy of S 2 × S 2. For any finite group G that acts freely on the threesphere we construct closed smooth fourmanifolds with fundamental group G which do not admit metrics of positive scalar curvature, but whose universal covers do admit such metrics. 1.
CLASSIFICATION OF SMOOTH EMBEDDINGS OF 3MANIFOLDS IN THE 6SPACE
, 2006
"... which often happens for 2m < 3n+4, then no concrete complete description of embeddings of nmanifolds into R m up to isotopy was known, except for disjoint unions of spheres. Let N be a closed connected orientable 3manifold. Our main result is the following description of the set Emb 6 (N) of embed ..."
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Cited by 3 (0 self)
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which often happens for 2m < 3n+4, then no concrete complete description of embeddings of nmanifolds into R m up to isotopy was known, except for disjoint unions of spheres. Let N be a closed connected orientable 3manifold. Our main result is the following description of the set Emb 6 (N) of embeddings N → R 6 up to isotopy. The Whitney invariant W: Emb 6 (N) → H1(N; Z) is surjective. For each u ∈ H1(N; Z) the Kreck invariant ηu: W −1 u → Z d(u) is bijective, where d(u) is the divisibility of the projection of u to H1(N; Z)/Tors. The group Emb 6 (S 3) is isomorphic to Z (Haefliger). This group acts on Emb 6 (N) by embedded connected sum. It was proved that the orbit space of this action maps under W (defined in a different way) bijectively to H1(N; Z) (by Vrabec and Haefliger’s smoothing theory). The new part of our classification result is determination of the orbits of the action. E. g. for N = RP 3 the action is free, while for N = S 1 × S 2 we construct explicitly an embedding f: N → R 6 such that for each knot l: S 3 → R 6 the embedding f#l is isotopic to f. Our proof uses new approaches involving the Kreck modified surgery theory or the BoéchatHaefliger formula for smoothing obstruction.
A CLASSIFICATION OF SMOOTH EMBEDDINGS OF 4MANIFOLDS IN 7SPACE, I
, 2008
"... Abstract. We work in the smooth category. Let N be a closed connected nmanifold and assume that m> n + 2. Denote by E m (N) the set of embeddings N → R m up to isotopy. The group E m (S n) acts on E m (N) by embedded connected summation of a manifold and a sphere. If E m (S n) is nonzero (which of ..."
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Cited by 1 (0 self)
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Abstract. We work in the smooth category. Let N be a closed connected nmanifold and assume that m> n + 2. Denote by E m (N) the set of embeddings N → R m up to isotopy. The group E m (S n) acts on E m (N) by embedded connected summation of a manifold and a sphere. If E m (S n) is nonzero (which often happens for 2m < 3n + 4) then until recently no results on this action and no complete description of E m (N) were known. Our main results are examples of the triviality and the effectiveness of this action, and a complete isotopy classification of embeddings into R 7 for certain 4manifolds N. The proofs use new approach based on the Kreck modified surgery theory and the construction of a new invariant. Corollary. (a) There is a unique embedding f: CP 2 → R 7 up to isoposition (i.e. for each two embeddings f, f ′ : CP 2 → R 7 there is a diffeomorphism h: R 7 → R 7 such that f ′ = h ◦ f). (b) For each embedding f: CP 2 → R 7 and each nontrivial embedding g: S 4 → R 7 the embedding f#g is isotopic to f. 1.
BETWEEN LOWER AND HIGHER DIMENSIONS (in the work of Terry Lawson)
"... There are several approaches to summarizing a mathematician’s research accomplishments, and each has its advantages and disadvantages. This article is based upon a talk given at Tulane that was aimed at a fairly general audience, including faculty members in other areas and graduate students who had ..."
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There are several approaches to summarizing a mathematician’s research accomplishments, and each has its advantages and disadvantages. This article is based upon a talk given at Tulane that was aimed at a fairly general audience, including faculty members in other areas and graduate students who had taken the usual entry level courses. As such, it is meant to be relatively nontechnical and to emphasize qualitative rather than quantitative issues; in keeping with this aim, references will be given for some standard topological notions that are not normally treated in entry level graduate courses. Since this was an hour talk, it was also not feasible to describe every single piece of published mathematical work that Terry Lawson has ever written; in particular, some papers like [42] and [50] would require lengthy digressions that are not easily related to the central themes in his main lines of research. Instead, we shall focus on some ways in which Terry’s work relates to an important thread in geometric topology; namely, the passage from studying problems in a given dimension to studying problems in the next dimensions. Qualitatively speaking, there are fairly welldeveloped theories for very low dimensions and for all sufficiently large dimensions, but between these ranges there are some dimensions in which the answers to many fundamental
SOME REMARKS ON GEOMETRIC SIMPLE CONNECTIVITY IN DIMENSION FOUR  Part A
, 2007
"... The present paper contains some complements and comments to the longer article Geometric simple connectivity in smooth four dimensional differential Topology, Part A, by the first author. Its aim is to be a useful companion when reading that article, and also to help in understand how it fits into t ..."
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The present paper contains some complements and comments to the longer article Geometric simple connectivity in smooth four dimensional differential Topology, Part A, by the first author. Its aim is to be a useful companion when reading that article, and also to help in understand how it fits into the first author’s program for the Poincaré conjecture.
1 ON SOME QUESTIONS OF FOUR DIMENSIONAL TOPOLOGY: A SURVEY OF MODERN RESEARCH
, 2005
"... Our physical intuition distinguishes four dimensions in a natural correspondence ..."
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Our physical intuition distinguishes four dimensions in a natural correspondence
ON RATIONAL HOMOTOPY OF FOURMANIFOLDS
, 2003
"... Abstract. We give explicit formulas for the ranks of the third and fourth homotopy groups of all oriented closed simply connected fourmanifolds in terms of their second Betti numbers. We also show that the rational homotopy type of these manifolds is classified by their rank and signature. 1. ..."
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Abstract. We give explicit formulas for the ranks of the third and fourth homotopy groups of all oriented closed simply connected fourmanifolds in terms of their second Betti numbers. We also show that the rational homotopy type of these manifolds is classified by their rank and signature. 1.