A Diffeomorphism Classification of Generalized Witten Manifolds

Abstract

We give a classification of a specific family of seven-dimensional manifolds, the generalized Witten manifolds up to homeomorphism and diffeomorphism. Using an approach suggested by J. Shaneson, we develop a modified surgery theory which fully classifies these manifolds. In contrast to previous approaches, this surgery theory is designed to be more readily applicable to higher dimensions. The family contains examples of manifolds which admit Einstein metrics and are homeomorphic but not diffeomorphic. These particular manifolds occur naturally in differential geometry and are of great interest to both differential geometers and physicists. Mathematics Subject Classification (2000). 55R15, 55R40, 57T35.

Torus actions on rationally elliptic manifolds

“…. , x [6,8,25] it has been shown that, already in dimension 7, there are infinitely many distinct homotopy types of such manifolds, distinguished by their cohomology rings.…”

“…The fact that, in each dimension, there are only finitely many rational homotopy types of manifolds (S 5 × m i=1 S 3 )/S 1 and ( m i=1 S 3 )/T 2 is in stark contrast to the general situation. Indeed, in [6,8,25] it has been shown that, already in dimension 7, there are infinitely many distinct homotopy types of such manifolds, distinguished by their cohomology rings.…”

Open manifolds with non-homeomorphic positively curved souls

“…where S 5 ⊂ C 3 , S 3 ⊂ C 2 , and k, l 1 , l 2 are nonzero integers such that k, l j are coprime for j = 1, 2; for such a space H 4 (M k,l1,l2 ) = Z l1l2 . We refer to [Esc05] for details.…”

“…Note that there are infinitely many manifolds in Theorems C and C that are not diffeomorphic to homogeneous spaces. Indeed, there are infinitely many spaces among Eschenburg and generalized Witten spaces that are not even homotopy equivalent to any homogeneous space [Sha02,Esc05].…”

Topological actions via gauge variations of higher structures

“…(ii) Generalized Witten manifolds N kl are defined as the total spaces of fiber bundles with fiber the lens space L k (ℓ 2 , ℓ 2 ) and structure group S 1 . They have H 4 (N kl ; Z) ∼ = Z |ℓ1ℓ2| [34].…”