Abstract
LetM n denote any closed connected CAT manifold of positive dimensionn. We define CATs(Mn) to be the smallest positive dimension of all closed connected CAT manifoldsN for which the CAT span ofM×N is strictly greater than the CAT span ofN. We determine a formula for this characteristic number which involves only the Kirby-Siebenmann numberks(M) ofM and a Stiefel-Whitney number. Several results on splitting the tangent bundles of closed 4-manifolds are obtained. For example, both the Euler number ofM 4 andks(M4) represent the total obstruction to positive CAT span for a non-smoothable closed connected 4-manifold.
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Dedicated to the memory of Professor Otto Endler
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Randall, D. On 4-manifolds and span-related numbers for cat manifolds. Manuscripta Math 69, 339–351 (1990). https://doi.org/10.1007/BF02567932
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DOI: https://doi.org/10.1007/BF02567932