An Introduction to Fibrewise Homotopy Theory

Crabb, M. C.; James, I. M.

doi:10.1007/978-1-4471-1265-5_1

Cited by 16 publications

(36 citation statements)

References 17 publications

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“…It now follows from the fibrewise obstruction theory (see [2,Proposition 2.15]) that the induced function between fiberwisehomotopy classes of maps over X…”

Section: Dimension and Category Estimatesmentioning

confidence: 99%

“…Let us denote by i g : F (g) → B and i f : F (f ) → A the fibrewise pointed homotopy fibres of the maps g and f . Moreover, the homotopy fibre of i g may be identified as j : Ω X (C) → F (g) where Ω X (C) is the fibrewise pointed loop space of C (see [2,Section I.13]). By the lifting property of homotopy fibres there are fibrewise pointed maps u, v such that the following diagram commutes:…”

Section: Weak Complexity Estimatesmentioning

confidence: 99%

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Spaces with high topological complexity

Franc

Pavešić

2014

Proceedings of the Royal Society of Edinburgh: Section A Mathem

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Abstract. By a formula of Farber [7, Theorem 5.2] the topological complexity TC(X) of a (p − 1)-connected, m-dimensional CW-complex X is bounded above by (2m + 1)/p + 1. There are also various lower estimates for TC(X) such as the nilpotency of the ring H * (X × X, ∆(X)), and the weak and stable topological compexity wTC(X) and σTC(X) (see [10]). In general the difference between these upper and lower bounds can be arbitrarily large. In this paper we investigate spaces whose topological complexity is close to the maximal value given by Farber's formula and show that in these cases the gap between the lower and upper bounds is narrow and that TC(X) often coincides with the lower bounds.

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“…It now follows from the fibrewise obstruction theory (see [2,Proposition 2.15]) that the induced function between fiberwisehomotopy classes of maps over X…”

Section: Dimension and Category Estimatesmentioning

confidence: 99%

Section: Weak Complexity Estimatesmentioning

confidence: 99%

Spaces with high topological complexity

Franc

Pavešić

2014

Proceedings of the Royal Society of Edinburgh: Section A Mathem

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“…For the notation and terminology we refer the reader to [5]. Throughout this section, V is a fixed real orthogonal n-dimensional representation of a compact Lie group G such that V e = ∅.…”

Section: Jfptamentioning

confidence: 99%

On relations between gradient and classical equivariant homotopy groups of spheres

Gęba

Izydorek

2012

J. Fixed Point Theory Appl.

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Abstract. We investigate relations between stable equivariant homotopy groups of spheres in classical and gradient categories. To this end, the auxiliary category of orthogonal equivariant maps, a natural enlargement of the category of gradient maps, is used. Our result allows for describing stable equivariant homotopy groups of spheres in the category of orthogonal maps in terms of classical stable equivariant groups of spheres with shifted stems. We conjecture that stable equivariant homotopy groups of spheres for orthogonal maps and for gradient maps are isomorphic.

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“…3 We write w(V ) = {α ∈ End(V ) | α * = α} (the space of self-adjoint endomorphisms of V ). If α ∈ w(V ) then the eigenvalues of α are real, so we can list them in descending order, repeated according to multiplicity.…”

Section: Symmentioning

confidence: 99%

“…3 The formal scheme (Ω X U (V )) E is the free commutative formal group over X E generated by the divisor DV .…”

mentioning

confidence: 99%

Common subbundles and intersections of divisors

Strickland

2002

Algebr. Geom. Topol.

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Let V 0 and V 1 be complex vector bundles over a space X . We use the theory of divisors on formal groups to give obstructions in generalised cohomology that vanish when V 0 and V 1 can be embedded in a bundle U in such a way that V 0 ∩ V 1 has dimension at least k everywhere. We study various algebraic universal examples related to this question, and show that they arise from the generalised cohomology of corresponding topological universal examples. This extends and reinterprets earlier work on degeneracy classes in ordinary cohomology or intersection theory.

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An Introduction to Fibrewise Homotopy Theory

Cited by 16 publications

References 17 publications

Spaces with high topological complexity

Spaces with high topological complexity

On relations between gradient and classical equivariant homotopy groups of spheres

Common subbundles and intersections of divisors

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