Abstract
Based on Morse theory for the energy functional on path spaces we develop a deformation theory for mapping spaces of spheres into orthogonal groups. This is used to show that these mapping spaces are weakly homotopy equivalent, in a stable range, to mapping spaces associated to orthogonal Clifford representations. Given an oriented Euclidean bundle V rightarrow X of rank divisible by four over a finite complex X we derive a stable decomposition result for vector bundles over the sphere bundle mathord {{mathbb {S}}}( mathord {{mathbb {R}}}oplus V) in terms of vector bundles and Clifford module bundles over X. After passing to topological K-theory these results imply classical Bott–Thom isomorphism theorems.
Highlights
In their seminal paper on Clifford modules Atiyah et al [2] describe a far-reaching interrelation between the representation theory of Clifford algebras and topological K-theory
Based on Morse theory for the energy functional on path spaces we develop a deformation theory for mapping spaces of spheres into orthogonal groups
This is used to show that these mapping spaces are weakly homotopy equivalent, in a stable range, to mapping spaces associated to orthogonal Clifford representations
Summary
In their seminal paper on Clifford modules Atiyah et al [2] describe a far-reaching interrelation between the representation theory of Clifford algebras and topological K-theory. Similar as before a vector bundle E → Vcan be constructed by a fiberwise clutching function along the “equator spheres” S(V ) in each fibre, and one may try to bring this clutching function into a favorable shape by a fiberwise deformation process similar as the one employed in [8] We will realize this program if V is oriented and of rank divisible by four in order to derive bundle theoretic versions of classical Bott–Thom isomorphism theorems in topological K-theory. This recovers Karoubi’s Clifford–Thom isomorphism theorem [9, Theorem IV.5.11] in the special case of oriented vector bundles V → X of rank divisible by four In this respect we provide a geometric approach to this important result, which is proven in [9] within the theory of Banach categories; see Discussion 10.10 at the end of our paper for more details. For completeness of the exposition we mention the analogous periodicity theorems for unitary and symplectic K-theory, which are in part difficult to find in the literature
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