Bundle Map Research Articles

On the distortion of conformal immersions between 2-space forms

Let 0~ and ~1 be 2-dimensional space forms (connected and complete Riemannian manifolds of class C 3 and of constant curvature K). We normalize K = 1, 0, 1, and speak of spherical, Euclidean, and hyperbolic space forms. Denote by O and 91 the metric tensors of ~ and ~ t . If f : ~ / -*~ is a conformal immersion and f , the induced linear map of the tangent bundles, then the distortion of f is the map

The tangent bundle of the long line

PROOF. The long line L is defined in Hocking and Young [1], for example. It is easy to check that L, with the topology defined there, supports the structure of a CX manifold. In fact, it is proved in [2] that L (there called the Alexandroff Half Line) can be made into a real analytic manifold (I thank the referee for this reference). Let us recall what it means for a real vector bundle E to be trivial. The bundle E2i-X is trivial if there is a bundle map j: E-*+n covering the identity: X-*X, where (n is the trivial Rn bundle over X and n is the dimension of the fibre of E. But this is easily seen to be equivalent to the existence of a bundle map f: E->Rn where Rn is considered as a bundle over a point *. (If w: {n-4Rn is projection onto the coordinate of each fibre, then f =w o q4 is such a map. If + is given then f-'Rn = in and Ei--hRn where f-' means the pull back under f.) So let TL be the tangent bundle of L and 0 be a Cx trivialization

The value of 𝐽 at a Samelson product

Introduction. At the Seattle Conference, 1963, M. G. Barratt asked the question, Can anything general be said about J: wk(SOn) -+7rk+n(Sn) on Samelson products? Lashof records [4] that Stasheff pointed out an answer to this question on the basis of results obtained by B. Steer in his Oxford thesis [8]. We offer here another answer, phrased in quite different terms, which is simple in both statement and proof. Let p: On+l--S be the usual bundle map with fiber On; in [5] we define the p-product: xr(O,) 0r8n,(Sn)-->ir,+8(Sn). The p-product of aEEirq(On) with O3=r8,(Sn) is denoted [a, ]3,. This product generalizes the J homomorphism: if Ln denotes the class of the identity map on Sn then J(a) [a, tn ], [5, Corollary 5.7].

The Realization of a Semisimplicial Bundle Map is a k-Bundle Map

The Realization of a Semisimplicial Bundle Map is a k-Bundle Map

The realization of a semisimplicial bundle map is a 𝑘-bundle map

0. Outline of main results. Let p: E -B be an ss fiber bundle map [2] with group r and fiber F. We will prove that the geometric realization [5] IPI: lEt IBI has a fiber-bundle like structure. The easiest way to describe it is to call it a fiber bundle in the category of by which we mean the following. A Hausdorff space X is a k-space if it has the weak topology with respect to its compact subsets. A mapf of a k-space X into any space Y is continuous if and only if its restriction to each compact subset is continuous. Each CW complex is a k-space. Let (k denote the category of k-spaces and continuous maps. It is easy to check that the category Ck iS closed under finite products. If X, Y are k-spaces, then the category Xx kY is the space retopologized with the weak topology with respect to compact subsets. Note that if X, Y are CW complexes, then Xx k Y is always a CW complex, the one whose cells are the products of those of X by those of Y, and whenever the space Xx Y is a CW complex, then Xx k Y= X x Y. Using this definition of category product, we can define in the usual way a kgroup (a group in the category (k), and the operation in (k of the k-group r on the k-space F. 0.1. DEFINITION. A map p: X -Y in (k is a coordinate k-bundle map (=fiber bundle in (k) if there is a covering of Y by the interiors of closed neighborhoods {Uj}, coordinate functions Oj: Uj x F -p l(Uj), transition functions gij: Ui nl Uj F, etc., as in Steenrod [6], except that group, operation, product always are meant in the sense of (k It is easy to see from [5] and the definition of k-product, etc., that for any ss complexes X, Y, that I Xx YI = Il x k I YI; that the realization of an ss group is a k-group; and that if the ss group G operates on F, that I GI operates on IFt in the category k Our interest is in bundle structures which are closely related to the given ss structure. For example:

Involutive Property of Resolutions of Differential Operators

Let E and E′ be C∞ vector bundles over a C∞ manifold M. Denote by Γ(E) (resp. by Γ(E′) the vector space of C∞ cross-sections of E (resp. of E′) over M. Take a linear differential operator of the first order D: Γ(E) → Γ(E′) induced by a vector bundle mapping σ(D): jl(E) ′ E′, where Jk(E) denotes the vector bundle of k-jets of cross-sections of E.

Cohomology and homology theories for categories of principal 𝐺-bundles

Introduction. In [4] G. W. Whitehead gives a method by which all extraordinary homology and cohomology theories may be generated by fundamental homotopy constructions. Unfortunately the definition of the extraordinary theories used is not sufficiently general so as to include all theories that would normally be considered as (co) homology theories. An example of a theory not included is (co) homology with local coefficients. In this paper, we will show how to extend the methods of Whitehead so as to include such theories. Let G be a fixed continuous group. If we consider the category of principal G-bundles over C.W. complexes, we may define a connected sequence of functors from this category to the category of abelian groups, satisfying the usual axioms of extraordinary (co) homology theory, with principal G-bundles replacing spaces, and bundle maps and homotopies replacing maps and homotopies. It is the purpose of this paper to show that such theories may be generated by homotopy constructions. In particular, we will indicate how local coefficient theory may be generated in this manner. In order to develop the homology theories spoken of above, it will first be necessary to define a class of spaces closely related to C.W. complexes but with just enough additional generality so as to include spaces for which a useful C.W. decomposition may not be available, but which are structured enough to give us a good deal of the type of information usually associated with a C.W.-decomposition. The definition of cohomology groups used in this paper was proposed by Professor Israel Berstein of Cornell University and will be exploited by him in a forthcoming paper on the Thom Isomorphism. This paper represents part of a doctoral dissertation written under Professor Berstein at Cornell University. 1. Piecewise-C.W. complexes. We will need three lemmas from homotopy theory. The first two will not be proved as they are quite easy. In the following,