Homotopically equivalent smooth manifolds. I

We study a smooth analogue of jumping curves of a holomorphic vector bundle, and use Yang–Mills theory over S2 to show that any non-trivial, smooth Hermitian vector bundle E over a smooth simply connected manifold, must have such curves. This is used to give new examples complex manifolds for which a non-trivial holomorphic vector bundle must have jumping curves in the classical sense (when c1(E) is zero). We also use this to give a new proof of a theorem of Gromov on the norm of curvature of unitary connections, and make the theorem slightly sharper. Lastly we define a sequence of new non-trivial integer invariants of smooth manifolds, connected to this theory of smooth jumping curves, and make some computations of these invariants. Our methods include an application of the recently developed Morse–Bott chain complex for the Yang–Mills functional over S2.

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Characteristic Classes and Smooth Manifolds

Springer Monographs in Mathematics - Topological Invariants of Stratified Spaces ◽

10.1007/3-540-38587-8_5 ◽

2007 ◽

pp. 99-122

Keyword(s):

Characteristic Classes ◽

Smooth Manifolds

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Averaging formula for Nielsen coincidence numbers

Nagoya Mathematical Journal ◽

10.1017/s0027763000009363 ◽

2007 ◽

Vol 186 ◽

pp. 69-93 ◽

Cited By ~ 8

Author(s):

Seung Won Kim ◽

Jong Bum Lee

Keyword(s):

Normal Subgroup ◽

Finite Index ◽

Holonomy Group ◽

Smooth Manifolds

AbstractIn this paper we study the averaging formula for Nielsen coincidence numbers of pairs of maps (f,g): M→N between closed smooth manifolds of the same dimension. Suppose that G is a normal subgroup of Π = π1(M) with finite index and H is a normal subgroup of Δ = π1(N) with finite index such that Then we investigate the conditions for which the following averaging formula holdswhere is any pair of fixed liftings of (f, g). We prove that the averaging formula holds when M and N are orientable infra-nilmanifolds of the same dimension, and when M = N is a non-orientable infra-nilmanifold with holonomy group ℤ2 and (f, g) admits a pair of liftings on the nil-covering of M.

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