Fibrewise Homotopy Theory

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

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

Cited by 89 publications

(124 citation statements)

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“…A fibrewise homotopy is a fibrewise map X × B I B → Y and we have a fibrewise homotopy equivalence in the obvious sense, which are the classical fibre homotopy and fibre homotopy equivalence, respectively. With this notion of fibrewise homotopies, we have a fibrewise fibration and a fibrewise cofibration, which are characterized by a fibrewise homotopy lifting property and a fibrewise homotopy extension property, respectively (see [3]). …”

Section: Fibrewise a N -Mapmentioning

confidence: 99%

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Splitting of gauge groups

Kishimoto

Kono

2010

Trans. Amer. Math. Soc.

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Abstract. Let G be a topological group and let P be a principal G-bundle over a based space B. We denote the gauge group of P by G(P ) and the based gauge group of P by G 0 (P ). Then the inclusion of the basepoint of B induces the exact sequence of topological groups 1 → G 0 (P ) → G(P ) → G → 1. We study the splitting of this exact sequence in the category of A n -spaces and A n -maps in connection with the triviality of the adjoint bundle of P and with the higher homotopy commutativity of G.

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Section: Fibrewise a N -Mapmentioning

confidence: 99%

“…equipped with an appropriate topology (see [3]), where Ω Y is the Moore path space of a space Y . Then the loop multiplication of Ω (π −1 (b)) makes Ω B X into a fibrewise topological monoid.…”

Section: Fibrewise a N -Mapmentioning

confidence: 99%

Splitting of gauge groups

Kishimoto

Kono

2010

Trans. Amer. Math. Soc.

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“…It is known that the cohomology Leray-Serre spectral sequence associated of the fibration U(r) [CJ98]). Therefore, f g (r) = P t (BU(r); Q) P t (U(r) 2g ; Q)…”

Section: 2mentioning

confidence: 99%

The Yang-Mills equations over Klein surfaces

Liu

2013

Journal of Topology

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Abstract. Moduli spaces of semi-stable real and quaternionic vector bundles of a fixed topological type admit a presentation as Lagrangian quotients and can be embedded into the symplectic quotient corresponding to the moduli variety of semi-stable holomorphic vector bundles of fixed rank and degree on a smooth complex projective curve. From the algebraic point of view, these Lagrangian quotients are connected sets of real points inside a complex moduli variety endowed with a real structure; when the rank and the degree are coprime, they are in fact the connected components of the fixed-point set of the real structure. This presentation as a quotient enables us to generalize the methods of Atiyah and Bott to a setting with involutions and compute the mod 2 Poincaré polynomials of these moduli spaces in the coprime case. We also compute the mod 2 Poincaré series of moduli stacks of all real and quaternionic vector bundles of a fixed topological type. As an application of our computations, we give new examples of maximal real algebraic varieties.

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“…We also make the technical restriction that fibrewise pointed spaces be homotopy well pointed (as in [9]). All fibrewise pointed spaces over B will be understood to be locally fibre homotopy trivial with each fibre of the homotopy type of a pointed CW complex.…”

Section: A Review Of Fibrewise Homology Theorymentioning

confidence: 99%