Obstruction theory on 7-manifolds
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ˇ Martin Cadek
© Springer-Verlag GmbH Germany, part of Springer Nature 2019
· Michael Crabb · Tomáš Salaˇc
Obstruction theory on 7-manifolds Received: 6 December 2018 / Accepted: 8 November 2019 Abstract. This paper gives a uniform, self-contained, and direct approach to a variety of obstruction-theoretic problems on manifolds of dimension 7 and 6. We give necessary and sufficient cohomological criteria for the existence of various G-structures on vector bundles over such manifolds especially using low dimensional representations of U(2).
1. Introduction Let M be a connected, closed, smooth, spinc manifold of dimension m = 6 or 7 and let ξ be an m-dimensional oriented real vector bundle over M admitting a spinc -structure. For various homomorphisms ρ : G → SO(m) from a compact Lie group G to SO(m) we consider the problem of reducing the structure group SO(m) of the vector bundle ξ to G via the representation ρ. Necessary and sufficient conditions for such reductions will be obtained in terms of the cohomology of M and cohomology characteristic classes of M and ξ . Thus as for methods and results the present paper is a continuation of [4]. Most of our results depend on the existence of 2-dimensional complex vector bundles over low dimensional manifolds. So we can provide more or less complete answers for all homomorphisms ρ which are connected with low dimensional representations of the group G = U(2). Our results complete the characterization of m-dimensional vector bundles over m-dimensional complexes (m = 6, 7) given in [13] and the results on the existence of vector fields over m-dimensional manifolds in [12]. The research of the third author was supported by the Grant 17-01171S of the Grant Agency of the Czech Republic. ˇ M. Cadek (B): Department of Mathematics, Masaryk University, Kotláˇrská 2, 611 37 Brno, Czech Republic. e-mail: [email protected] M. Crabb: Institute of Mathematics, University of Aberdeen, Aberdeen AB24 3UE, UK e-mail: [email protected] T. Salaˇc: Faculty of Mathematics and Physics, Charles University, Sokolovská, 83 186 75 Prague, Czech Republic e-mail: [email protected] Mathematics Subject Classification: 55R25 · 55R40 · 55S35
https://doi.org/10.1007/s00229-019-01165-2
ˇ M. Cadek et al.
We conclude this introduction by describing our results for 7-manifolds in the case that G is the group Sp(1) and the manifold M and vector bundle ξ are spin. There are 4 irreducible real Sp(1)-modules of dimension at most 7: the Lie algebra A1 of dimension 3, the defining 4-dimensional module E (= H), a module A2 of dimension 5, and a module A3 of dimension 7. We thus have, up to equivalence, the following seven 7-dimensional real Sp(1)-modules and associated representations ρ: (i) R7 , (ii) E ⊕ R3 , (iii) A1 ⊕ R4 , (iv) E ⊕ A1 , (v) A1 ⊕ A1 ⊕ R, (vi) A2 ⊕ R2 , (vii) A3 . Theorem 1.1. Let ξ be a 7-dimensional vector bundle with w1 ξ = 0 and w2 ξ = 0 over a 7-dimensional spin manifold M. Then the structure group of ξ reduces from Spin(7) to Sp(1) through ρ if and only if the spin characteris
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