The rational homotopy type of (n-1)-connected manifolds of dimension up to 5n-3

We define the Bianchi-Massey tensor of a topological space X to be a linear map from a subquotient of the fourth tensor power of H*(X). We then prove that if M is a closed (n-1)-connected manifold of dimension at most 5n-3 (and n > 1) then its rational homotopy type is determined by its cohomology algebra and Bianchi-Massey tensor, and that M is formal if and only if the Bianchi-Massey tensor vanishes. We use the Bianchi-Massey tensor to show that there are many (n-1)-connected (4n-1)-manifolds that are not formal but have no non-zero Massey products, and to present a classification of simply-connected 7-manifolds up to finite ambiguity.Comment: 31 pages. v3: Corrected statement of Theorem 1.5. To appear in the Journal of Topolog

The classification of 2-connected 7-manifolds

We present a classification theorem for closed smooth spin 2-connected 7-manifolds M. This builds on the almost-smooth classification from the first author's thesis. The main additional ingredient is an extension of the Eells-Kuiper invariant for any closed spin 7-manifold, regardless of whether the spin characteristic class p_M in the fourth integral cohomology of M is torsion. In addition we determine the inertia group of 2-connected M - equivalently the number of oriented smooth structures on the underlying topological manifold - in terms of p_M and the torsion linking form.Comment: Corrected the definition of pseudo-isotopy of almost diffeomorphisms, 56 pages. To appear in Proc. Lond. Math. So

New invariants of G_2-structures

We define a Z/48-valued homotopy invariant nu of a G_2-structure on the tangent bundle of a closed 7-manifold in terms of the signature and Euler characteristic of a coboundary with a Spin(7)-structure. For manifolds of holonomy G_2 obtained by the twisted connected sum construction, the associated torsion-free G_2-structure always has nu = 24. Some holonomy G_2 examples constructed by Joyce by desingularising orbifolds have odd nu. We define a further homotopy invariant xi of G_2-structures such that if M is 2-connected then the pair (nu, xi) determines a G_2-structure up to homotopy and diffeomorphism. The class of a G_2-structure is determined by nu on its own when the greatest divisor of p_1(M) modulo torsion divides 224; this sufficient condition holds for many twisted connected sum G_2-manifolds. We also prove that the parametric h-principle holds for coclosed G_2-structures.Comment: 26 pages, 1 figure. v3: Defined further invariant, strengthened classification results, changed titl

Finite group actions on Kervaire manifolds

The (4k+2)-dimensional Kervaire manifold is a closed, piecewise linear (PL) manifold with Kervaire invariant 1 and the same homology as the product of two (2k+1)-dimensional spheres. We show that a finite group of odd order acts freely on a Kervaire manifold if and only if it acts freely on the corresponding product of spheres. If the Kervaire manifold M is smoothable, then each smooth structure on M admits a free smooth involution. If k + 1 is not a 2-power, then the Kervaire manifold in dimension 4k+2 does not admit any free TOP involutions. Free "exotic" (PL) involutions are constructed on the Kervaire manifolds of dimensions 30, 62, and 126. Each smooth structure on the 30-dimensional Kervaire manifold admits a free Z/2 x Z/2 action.Comment: 36 pages. Final version: Advances in Mathematics (to appear

Functorial semi-norms on singular homology and (in)flexible manifolds

A functorial semi-norm on singular homology is a collection of semi-norms on the singular homology groups of spaces such that continuous maps between spaces induce norm-decreasing maps in homology. Functorial semi-norms can be used to give constraints on the possible mapping degrees of maps between oriented manifolds. In this paper, we use information about the degrees of maps between manifolds to construct new functorial semi-norms with interesting properties. In particular, we answer a question of Gromov by providing a functorial semi-norm that takes finite positive values on homology classes of certain simply connected spaces. Our construction relies on the existence of simply connected manifolds that are inflexible in the sense that all their self-maps have degree -1, 0, or 1. The existence of such manifolds was first established by Arkowitz and Lupton; we extend their methods to produce a wide variety of such manifolds.Comment: 37 pages, 1 figure; v2: added some references, corrected some typos; v3: added observation on multiplicative finite functorial semi-norms; v4: corrected a mistake in Corollary 3.2 (the main results are not affected); to appear in AG

Positive Ricci curvature on highly connected manifolds

For $k \ge 2,$ let $M^{4k-1}$ be a $(2k{-}2)$-connected closed manifold. If $k \equiv 1$ mod $4$ assume further that $M$ is $(2k{-}1)$-parallelisable. Then there is a homotopy sphere $\Sigma^{4k-1}$ such that $M \sharp \Sigma$ admits a Ricci positive metric. This follows from a new description of these manifolds as the boundaries of explicit plumbings.Comment: Corrected some minor typos and changed document class to amsart. The new document class added 10 pages, so the paper is now now 46 page