Codimension two souls and cancellation phenomena

Abstract

For each m ≥ 0 we find an open (4m + 9) -dimensional simply connected manifold with complete nonnegatively curved metrics whose souls are nondiffeomorphic, homeomorphic, and have codimension 2 . We give a diffeomorphism classification of the pairs (N, soul) when N is the total space of a nontrivial complex line bundle over S 7 × CP 2 ; up to diffeomorphism there are precisely three such pairs, distinguished by their nondiffeomorphic souls.

Tangential thickness of manifolds

“…, 27. It is proved in [14] that although Σ 7 (d) × CP 2 falls into one tangential homotopy type, there are four oriented (three unoriented) diffeomorphism classes of these manifolds, each admitting a nonnegatively curved metric by the main result of [32]. Similar results hold for manifolds of the form Σ 7 × CP 2n for all n such that n ≡ 0 mod 3 [76].…”

“…Finally, Section 8.1 contains some comments and remarks concerning smooth tangential thickness. Some of the techniques and ideas of this paper were applied in [14] and [15] when studying and classifying open complete manifolds of nonnegative curvature (see also [60] for further results on such questions). By the results of J. Cheeger and D. Gromoll [21], such manifolds are diffeomorphic to the total space of a normal bundle to a compact locally geodesic submanifold called a soul.…”

Open manifolds with non-homeomorphic positively curved souls

“…All we know is that any pair of souls as in Theorem A necessarily has codimension at least three: according to [BKS11], any two codimension-two souls of a simply connected open manifold are homeomorphic. There is, however, the following result on positively-curved codimension-two souls due to Belegradek, Kwasik and Schultz: Indeed, this is essentially the case m = 0 of Theorem 1.4 in [BKS15]; the exact statement may easily be extracted from the proof of this theorem given there (see page 41). This result does not rely on the "Work Horse Theorem" stated as Theorem 15 above.…”

“…metrics (see [15] and the recent preprint [12]), and they all become diffeomorphic after taking the product with R 3 . Thus, the obvious product metrics yield fifteen non-diffeomorphic souls of S 7 × R 3 . In a more elaborate construction, Belegradek showed that S 3 × S 4 × R 5 admits infinitely many souls that are pairwise non-homeomorphic [1].…”

Space of nonnegatively curved metrics and pseudoisotopies

“…Let M K≥0 (V ) be the associated moduli space, the quotient space of R K≥0 (V ) by the above Diff V action. Many open manifolds V for which M K≥0 (V ) is not path-connected, or even has infinitely many path-components, were found in [KPT05,BKS11,BKS15,Otta]. On the other hand, it was shown in [BH15] that R K≥0 (R 2 ) is homeomorphic to the separable Hilbert space, and the associated moduli space M K≥0 (R 2 ) cannot be separated by a closed subset of finite covering dimension.…”