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The Convenient Setting of Global Analysis
, 1997
"... ichor i Table of Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 CHAPTER I Calculus of Smooth Mappings . . . . . . . . . . . . . . . . . . . . 4 1. Smooth Curves . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1a. Completeness . . . . . . . . . . . . . . ..."
Abstract

Cited by 198 (47 self)
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ichor i Table of Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 CHAPTER I Calculus of Smooth Mappings . . . . . . . . . . . . . . . . . . . . 4 1. Smooth Curves . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1a. Completeness . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1b. Smooth Mappings and the Exponential Law . . . . . . . . . . . . . 17 2. The c 1 Topology . . . . . . . . . . . . . . . . . . . . . . . . . 29 3. Uniform Boundedness Principles and Multilinearity . . . . . . . . . . 47 3a. Some Spaces of Smooth Functions . . . . . . . . . . . . . . . . . 59 Historical remarks on the development of smooth calculus . . . . . . . . . 63 CHAPTER II Partitions of Unity . . . . . . . . . . . . . . . . . . . . . . . . . 68 5. D
Natural Operations In Differential Geometry
, 1993
"... CONTENTS PREFACE . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 CHAPTER I. MANIFOLDS AND LIE GROUPS . . . . . . . . . . . . . . . . 4 1. Differentiable manifolds . . . . . . . . . . . . . . . . . . . . . 4 2. Submersions and immersions . . . . . . . . . . . . . . . . . . 11 3. Vector ..."
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Cited by 55 (17 self)
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CONTENTS PREFACE . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 CHAPTER I. MANIFOLDS AND LIE GROUPS . . . . . . . . . . . . . . . . 4 1. Differentiable manifolds . . . . . . . . . . . . . . . . . . . . . 4 2. Submersions and immersions . . . . . . . . . . . . . . . . . . 11 3. Vector fields and flows . . . . . . . . . . . . . . . . . . . . . 16 4. Lie groups . . . . . . . . . . . . . . . . . . . . . . . . . . 30 5. Lie subgroups and homogeneous spaces . . . . . . . . . . . . . 41 CHAPTER II. DIFFERENTIAL FORMS . . . . . . . . . . . . . . . . . . . 49 6. Vector bundles . . . . . . . . . . . . . . . . . . . . . . . . 49 7. Differential forms . . . . . . . . . . . . . . . . . . . . . . . 61 8. Derivations on the algebra of differential forms and the FrolicherNijenhuis bracket . . . . . . . . . . . . .
Higher order intersection numbers of 2spheres in 4manifolds
 ALGEBRAIC & GEOMETRIC TOPOLOGY
, 2000
"... This is the beginning of an obstruction theory for deciding whether a map f: S2 → X is homotopic to a topologically flat embedding, in the presence of fundamental group and in the absence of dual spheres in a topological 4manifold X. The first obstruction is Wall’s well known selfintersection nu ..."
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Cited by 16 (9 self)
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This is the beginning of an obstruction theory for deciding whether a map f: S2 → X is homotopic to a topologically flat embedding, in the presence of fundamental group and in the absence of dual spheres in a topological 4manifold X. The first obstruction is Wall’s well known selfintersection number µ(f) which tells the whole story in higher dimensions. Our second order obstruction τ(f) is defined if µ(f) vanishes and has formally very similar properties, except that it lies in a quotient of the group ring of two copies of π1X modulo S3symmetry (rather then just one copy modulo S2symmetry). It generalizes to the nonsimply connected setting the KervaireMilnor invariant defined in [2] and [12] which corresponds to the Arfinvariant of knots in 3space. We also give necessary and sufficient conditions for homotoping three maps f1, f2, f3: S2 → X to a position in which they have disjoint images. The obstruction λ(f1, f2, f3) generalizes Wall’s intersection number λ(f1, f2) which answers the same question for two spheres but is not sufficient (in dimension 4) for three spheres. In the same way as intersection numbers correspond to linking numbers in 3space, our new invariant corresponds to the Milnor invariant µ(1, 2, 3), generalizing the Matsumoto triple to the non simplyconnected setting. Finally, we explain some simple algebraic properties of these new cubic forms on π2(X) in Theorem 3. These are straightforward generalizations of the properties of quadratic forms as defined by Wall [14, §5]. A particularly attractive formula is λ(f, f, f) = ∑ τ(f) σ σ∈S3 which generalizes the well known fact that Wall’s invariants satisfy λ(f, f) = µ(f) + µ(f) = ∑ µ(f) σ for an immersion f with trivial normal bundle. σ∈S2 1.
Differential 3knots in 5space with and without selfintersections
 Topology 40 (2001), 157–196; MR1791271 (2001h:57033
"... Abstract. Regular homotopy classes of immersions S 3 → R 5 constitute an infinite cyclic group. The classes containing embeddings form a subgroup of index 24. The obstruction for a generic immersion to be regularly homotopic to an embedding is described in terms of geometric invariants of its self i ..."
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Cited by 12 (4 self)
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Abstract. Regular homotopy classes of immersions S 3 → R 5 constitute an infinite cyclic group. The classes containing embeddings form a subgroup of index 24. The obstruction for a generic immersion to be regularly homotopic to an embedding is described in terms of geometric invariants of its self intersection. Geometric properties of self intersections are used to construct two invariants J and St of generic immersions which are analogous to Arnold’s invariants of plane curves [1]. We prove that J and St are independent first order invariants and that any first order invariant is a linear combination of these. As byproducts, some invariants of immersions S 3 → R 4 are obtained. Using them, we find restrictions on the topology of self intersections. 1.
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.
Quantifying transversality by measuring the robustness of intersections
 Manuscript, Dept. Comput. Sci., Duke Univ
, 2009
"... By definition, transverse intersections are stable under infinitesimal perturbations. Using persistent homology, we extend this notion to sizeable perturbations. Specifically, we assign to each homology class of the intersection its robustness, the magnitude of a perturbation necessary to kill it, a ..."
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Cited by 11 (7 self)
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By definition, transverse intersections are stable under infinitesimal perturbations. Using persistent homology, we extend this notion to sizeable perturbations. Specifically, we assign to each homology class of the intersection its robustness, the magnitude of a perturbation necessary to kill it, and prove that robustness is stable. Among the applications of this result is a stable notion of robustness for fixed points of continuous mappings and a statement of stability for contours of smooth mappings.
Totally skew embeddings of manifolds
"... Abstract. We study a version of Whitney’s embedding problem in projective geometry: What is the smallest dimension of an affine space that can contain an ndimensional submanifold without any pairs of parallel or intersecting tangent lines at distinct points? This problem is related to the generaliz ..."
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Cited by 10 (8 self)
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Abstract. We study a version of Whitney’s embedding problem in projective geometry: What is the smallest dimension of an affine space that can contain an ndimensional submanifold without any pairs of parallel or intersecting tangent lines at distinct points? This problem is related to the generalized vector field problem, existence of nonsingular bilinear maps, and the immersion problem for real projective spaces. We use these connections and other methods to obtain several specific and general bounds for the desired dimension. 1.
van Kampen’s embedding obstructions for discrete groups
 Invent. Math
, 2002
"... We give a lower bound to the dimension of a contractible manifold on which a given group can act properly discontinuously. In particular, we show that the nfold product of nonabelian free groups cannot act properly discontinuously on R 2n−1. 1 ..."
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Cited by 9 (2 self)
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We give a lower bound to the dimension of a contractible manifold on which a given group can act properly discontinuously. In particular, we show that the nfold product of nonabelian free groups cannot act properly discontinuously on R 2n−1. 1
NONISOTOPIC LEGENDRIAN SUBMANIFOLDS IN R 2n+1
"... In the standard contact (2n +1)spacewhenn>1, we construct infinite families of pairwise nonLegendrian isotopic, Legendrian nspheres, ntori and surfaces which are indistinguishable using classically known invariants. When n is even, these are the first known examples of nonLegendrian isotopic, L ..."
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Cited by 8 (3 self)
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In the standard contact (2n +1)spacewhenn>1, we construct infinite families of pairwise nonLegendrian isotopic, Legendrian nspheres, ntori and surfaces which are indistinguishable using classically known invariants. When n is even, these are the first known examples of nonLegendrian isotopic, Legendrian submanifolds of (2n + 1)space. Such constructions indicate a rich theory of Legendrian submanifolds. To distinguish our examples, we compute their contact homology which was rigorously defined in this situation in [7]. 1.