Some results on extension of maps and applications (original) (raw)
Obstructions to fibering a manifold
Geometriae Dedicata, 2010
Given a map f : M → N of closed topological manifolds we define torsion obstructions whose vanishing is a necessary condition for f being homotopy equivalent to a projection of a locally trivial fiber bundle. If N = S 1 , these torsion obstructions are identified with the ones due to Farrell .
Transactions of the American Mathematical Society, 1966
This paper is the natural sequel to Geometry of immersions. I [4]. The principal aim of the two papers as a whole is to sufficiently alter the structure of the theory of singularities of smooth maps of Thom and Whitney, so as to be applicable to some problems concerning geometrically defined singularities, which depend upon particular choices of affine or Riemannian connections. Then having so altered this branch of differential topology, to apply our general results to some specific problems. In [4] we developed an analog of the Thom-Whitney transversality theory, which was applicable to pth order osculating maps defined with respect to a given set of "generalized connections." We then used the transversality theory to prove pth order analogs to the embedding, immersion, isotopy and regular homotopy theorems of Whitney (see [16]). Having developed this analog of the transversality theory, it is natural to see whether we can prove an analog of Thorn's polynomial theorem (see Séminaire Cartan 56-57, Exposé 8). Part I of this paper deals with this question. It consists in formally extending the concept of a singularity of a homomorphism between vector bundles to take into account reductions of structural groups which will be given by "geometric" structures on our manifolds. With these geometric reductions built into the theory, a very general existence theorem of the Thom type is stated and proved (Theorems 2.5 and 2.6). The proof is a reworking with more details of the basic ideas in Thorn's proof as presented by Haefliger in the Cartan seminar. (See Exposé 8, Séminaire Cartan 56-57). However some pains are taken to insure that the gaps in the earlier presentation (i.e., the whole question of fundamental classes) are not present in this one. With the recent work of Borel, Moore, and Haefliger, (see [1] and [2]),
On the Extension of Certain Maps with Values in Spheres
Bulletin of the Polish Academy of Sciences Mathematics, 2008
Let E be an oriented, smooth and closed m-dimensional manifold with m ≥ 2 and V ⊂ E an oriented, connected, smooth and closed (m − 2)-dimensional submanifold which is homologous to zero in E. Let S n−2 ⊂ S n be the standard inclusion, where S n is the n-sphere and n ≥ 3. We prove the following extension result: if h : V → S n−2 is a smooth map, then h extends to a smooth map g : E → S n transverse to S n−2 and with g −1 (S n−2) = V. Using this result, we give a new and simpler proof of a theorem of Carlos Biasi related to the ambiental bordism question, which asks whether, given a smooth closed n-dimensional manifold E and a smooth closed m-dimensional submanifold V ⊂ E, one can find a compact smooth (m + 1)-dimensional submanifold W ⊂ E such that the boundary of W is V .
About the extension problem for proper maps
Topology and its Applications, 1987
In this paper, we study the extension problem in the category of topological spaces and proper maps. To attack this problem a new proper cohomology theory and a new obstruction cocycle are defined. This cohomology theory has coefficients in a morphism r'+ n where r' is a pro-abelian group and n is an abelian group. Let K" be the n-skeleton of a second countable, locally compact cell complex K, and let Y be a topological space with a colinal sequence of compact subsets 0 = M,, c M, c M2 c. .. c Y such that Y-M, is a path-connected n-simple space. In this case, the sequence. ..-.~"(Y-M*)~P"(Y-,)~~"(Y) can be seen as a morphism of the pro-abelian group r' = {rm(Y-M,)li 2 1) to the abelian group ?r = r,,(Y). Then we define an obstruction cocycle c"+' (g) with coefficient in v'+ r and prove the following results. (Proposition) A proper map g:K" + Y has a proper extension over K"+' if and only if _c""(g) = 0. (Theorem) Let g : K" + Y be a proper map. Then g/K"-' can be properly extended over K "+' if and only if _c"+' (g) is cohomologous to zero. AMS(MOS) Subj. Class.: 55836 proper homotopy proper obstruction cocycle proper extensions proper difference cochain proper cohomology
2001
The algebraic theory of surgery gives a necessary and suffcient chain level condition for a space with n-dimensional Poincare duality to be homotopy equivalent to an n-dimensional topological manifold. A relative version gives a necessary and suffcient chain level condition for a simple homotopy equivalence of n-dimensional topological manifolds to be homotopic to a homeomorphism. The chain level obstructions come from a chain level interpretation of the fibre of the assembly map in surgery.
Noninvertibility and``Semi-''Analogs of (Super) Manifolds, Fiber Bundles and Homotopies
1996
Supersymmetry contains initially noninvertible objects, but it is common to deal with the invertible ones only, factorizing former in some extent. We propose to reconsider this ansatz and try to redefine such fundamental notions as supermanifolds, fiber bundles and homotopies using some weakening invertibility conditions. The prefix semireflects the fact that the underlying morphisms form corresponding semigroups consisting of a known group part and a new ideal noninvertible part. We found that the absence of invertibility gives us the generalization of the cocycle conditions for transition functions of supermanifolds and fiber bundles in a natural way, which can lead to construction of noninvertible analogs ofČech cocycles. We define semi-homotopies, which can be noninvertible and describe mappings into the semi-supermanifolds introduced. * Alexander von Humboldt Fellow
Transactions of the American Mathematical Society, 1965
Introduction. For the past twenty-five years much of global differential geometry and topology has centered around what we now call differential topology. Much emphasis has been placed on those mathemathical ideas (vector bundles, their characteristic classes, and general position arguments) which depend only on the differentiable structure of manifolds and maps. In particular the work of Whitney and Thorn (see Levine [1] and Whitney [l]-[6]) belong to that domain. Our principal aim in this paper is to show that the tools of differential topology can be suitably modified to become applicable to differential geometry. Our second goal is to use these tools to investigate some "higher-order" problems in differential geometry (that is, problems involving derivatives of order greater than 1). We will assume that all manifolds discussed in this introduction are C°° and satisfy the second axiom of countability. Furthermore we will use the words smooth, differentiable, and C00 interchangeably. The first problem encountered in such a program, is the one of finding the "natural" objects on which to build the theory. Of utmost importance is the fact that the theory of linear connections, central to differential geometry, must be built into these objects in a relatively meaningful way. The best objects that we found for this task were Pohl's (see Pohl [1]) pth order tangent bundles TP(X), where X is a smooth manifold. The first section contains the definition of TP(Z) and a résumé of their important properties. One of the important facts is the fact that the TpiX) satisfy the following short exact sequences of vector bundles: (£,)0 -TpiX) -> Tp+ yiX) -+ 0 p+1TyiX) -> 0, where 0P denotes the p-fold symmetric tensor product of vector bundles. Of equal importance is the fact that iff : X -* Y is a smooth map there is induced a vector bundle homomorphism Tpif): TpiX)-+TpiY) covering /. Probably the most important aspect of this approach is that some connection theory can be built into it. It is a classical result (see Ambrose, Singer, and Palais [1]) that splittings of the exact sequence 0-> TyiX)-+ T2iX)^02TyiX)-+0 are in 1-1 correspondence with symmetric linear connections on X. It is not hard
Some results on vector bundle monomorphisms
Algebraic & Geometric Topology, 2007
In this paper we use the singularity method of Koschorke [2] to study the question of how many different nonstable homotopy classes of monomorphisms of vector bundles lie in a stable class and the percentage of stable monomorphisms which are not homotopic to stabilized nonstable monomorphisms. Particular attention is paid to tangent vector fields. This work complements some results of Koschorke [3; 4], Libardi-Rossini [7] and Libardi-do Nascimento-Rossini [6].