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Register a new userThe rational homotopy type of (n-1)-connected manifolds of dimension up to 5n-3
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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.
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Building on work of Stolz, we prove for integers $0 le d le 3$ and $k>232$ that the boundaries of $(k-1)$-connected, almost closed $(2k+d)$-manifolds also bound parallelizable manifolds. Away from finitely many dimensions, this settles longstanding questions of C.T.C. Wall, determines all Stein fillable homotopy spheres, and proves a conjecture of Galatius and Randal-Williams. Implications are drawn for both the classification of highly connected manifolds and, via work of Kreck and Krannich, the calculation of their mapping class groups. Our technique is to recast the Galatius and Randal-Williams conjecture in terms of the vanishing of a certain Toda bracket, and then to analyze this Toda bracket by bounding its $mathrm{H}mathbb{F}_p$-Adams filtrations for all primes $p$. We additionally prove new vanishing lines in the $mathrm{H}mathbb{F}_p$-Adams spectral sequences of spheres and Moore spectra, which are likely to be of independent interest. Several of these vanishing lines rely on an Appendix by Robert Burklund, which answers a question of Mathew about vanishing curves in $mathrm{BP} langle n rangle$-based Adams spectral sequences.
We prove that that the homotopy type of the path connected component of the identity in the contactomorphism group is characterized by the homotopy type of the diffeomorphism group plus some data provided by the topology of the formal contactomorphism space. As a consequence, we show that every connected component of the space of Legendrian long knots in $R^3$ has the homotopy type of the corresponding smooth long knot space. This implies that any connected component of the space of Legendrian embeddings in $NS^3$ is homotopy equivalent to the space $K(G,1)timesU(2)$, with $G$ computed by A. Hatcher and R. Budney. Similar statements are proven for Legendrian embeddings in $R^3$ and for transverse embeddings in $NS^3$. Finally, we compute the homotopy type of the contactomorphisms of several tight $3$-folds: $NS^1 times NS^2$, Legendrian fibrations over compact orientable surfaces and finite quotients of the standard $3$-sphere. In fact, the computations show that the method works whenever we have knowledge of the topology of the diffeomorphism group. We prove several statements on the way that have interest by themselves: the computation of the homotopy groups of the space of non-parametrized Legendrians, a multiparametric convex surface theory, a characterization of formal Legendrian simplicity in terms of the space of tight contact structures on the complement of a Legendrian, the existence of common trivializations for multi-parametric families of tight $R^3$, etc.
Bredon has constructed a 2-dimensional compact cohomology manifold which is not homologically locally connected, with respect to the singular homology. In the present paper we construct infinitely many such examples (which are in addition metrizable spaces) in all remaining dimensions $n ge 3$.
Let $G$ be a simply-connected simple compact Lie group and let $M$ be an orientable smooth closed 4-manifold. In this paper we calculate the homotopy type of the suspension of $M$ and the homotopy types of the gauge groups of principal $G$-bundles over $M$ when $pi_1(M)$ is: (1)~$mathbb{Z}^{*m}$, (2)~$mathbb{Z}/p^rmathbb{Z}$, or (3)~$mathbb{Z}^{*m}*(*^n_{j=1}mathbb{Z}/p_j^{r_j}mathbb{Z})$, where $p$ and the $p_j$s are odd primes.
We determine the number of distinct fibre homotopy types for the gauge groups of principal $Sp(2)$-bundles over a closed, simply-connected four-manifold.
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