Some results on extension of maps and applications (original) (raw)
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Contributions to the homotopy theory of mapping spaces
1972
Spaces of continuous m~s Preliminaries to Part I 15-21 15 §1 Basic definitions " 2 Evaluation fibrations defined by a space of mappf.ng e with a suspension as domain. §3 A :rundanental t.re or-em of O.W.Whitehead and some facts about \'lhitehead products in spheres .. • 19 17 Chapter 2: Evaluation fibrat10ns §1 Sect ions in evalua tion fibrations••• .... 22-34 .. 22 §2 The structure of a neutral evaluation fibration 24 §3 Fibre homotopy equivalence of evaluation fibrations. . 28 §4 Strong fibre homotopy equivalence of evaluation fibrations 31 Chapter 3: Homotopy equivalence of components in spaces of maps between spheres,. Tte order of ?t n _ l (Go. (s", Sn» for n even and a. E 7t n (Sn) §2 Characterization of tbe homotopy type of the. 35-42 §1 .. 35 neutral component. 37 §3 The division int 0 homotopy types of the components in the mapping spaces G~S.r",Sn) and G(sn+l,Sn) ....
Making lifting obstructions explicit
Proceedings of the London Mathematical Society, 2013
If P → X is a topological principal K-bundle and K a central extension of K by Z, then there is a natural obstruction class δ1(P ) ∈Ȟ 2 (X, Z) in sheaf cohomology whose vanishing is equivalent to the existence of a K-bundle P over X with P ∼ = P /Z. In this paper we establish a link between homotopy theoretic data and the obstruction class δ1(P ) which in many cases can be used to calculate this class in explicit terms. Writing ∂ P d : π d (X) → π d−1 (K) for the connecting maps in the long exact homotopy sequence, two of our main results can be formulated as follows. If Z is a quotient of a contractible group by the discrete group Γ, then the homomorphism π3(X) → Γ induced by δ1(P ) ∈Ȟ 2 (X, Z) ∼ = H 3 sing (X, Γ) coincides with ∂ K 2 •∂ P 3 and if Z is discrete, then δ1(P ) ∈Ȟ 2 (X, Z) induces the homomorphism −∂ K 1 •∂ P
A note on nonstable monomorphisms¶of vector bundles
manuscripta mathematica, 2000
In this paper we consider the question of the existence of a nonstable vector bundle monomorphism u : α → β over a closed, connected and smooth manifold M, when dimension of α = 3, dimension of β = dimension of M = n ≡ 0(4). The singularity method provides the full obstruction to this problem and under some homological hypothesis we can compute it in terms of well known invariants. an auxiliary virtual vector bundle over M. We can assume (because of the dimension hypothesis) that there is a generic vector bundle homomorphism u : α → β which has rank at least two in each point of M. The singularity is the set S := { x ∈ M | rank(u x : α x → β x) = 2}.
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