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

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 Frolicher-Nijenhuis bracket . . . . . . . . . . . . .

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Abstract. 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 4-manifold X. The first obstruction is Wall’s well known self-intersection 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 S3-symmetry (rather then just one copy modulo S2-symmetry). It generalizes to the non-simply connected setting the Kervaire-Milnor invariant defined in [2] and [12] which corresponds to the Arf-invariant of knots in 3-space. 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 3-space, our new invariant corresponds to the Milnor invariant µ(1, 2, 3), generalizing the Matsumoto triple to the non simply-connected 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.

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 by-products, some invariants of immersions S 3 → R 4 are obtained. Using them, we find restrictions on the topology of self intersections. 1.

Abstract. A clear understanding of topology of higher-dimensional objects is important in many branches of both pure and applied mathematics. In this survey we attempt to present some results of higher-dimensional 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 non-specialists. We outline a general approach to embeddings via the classical van Kampen-Shapiro-Wu-Haefliger-Weber ’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 Haefliger-Weber embedding theorem below the metastable dimension range and examples showing that other analogues of this theorem are false outside the metastable dimension range. 1.

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 n-dimensional 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 non-singular 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.

Abstract. For a given n-dimensional manifold M n we study the problem of finding the smallest integer N(M n) such that M n admits a smooth embedding in the Euclidean space R N without intersecting tangent spaces. We use the Poincaré-Hopf index theorem to prove that N(S 1) = 4, and construct explicit examples to show that N(S n) ≤ 3n + 3, where S n denotes the n-sphere. Finally, for any closed manifold M n, we show that 2n + 1 ≤ N(M n) ≤ 4n + 1. 1.

In the standard contact (2n +1)-spacewhenn>1, we construct infinite families of pairwise non-Legendrian isotopic, Legendrian n-spheres, n-tori and surfaces which are indistinguishable using classically known invariants. When n is even, these are the first known examples of non-Legendrian 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.

Abstract. Quadratically parametrized smooth maps from one complex projective space to another are constructed as projections of the Segre map of the complexification. A classification theorem relates equivalence classes of projections to congruence classes of matrix pencils. Maps from the 2-sphere to the complex projective plane, which generalize stereographic projection, and immersions of the complex projective plane in four and five complex dimensions, are considered in detail. Of particular interest are the CR singular points in the image.