Homotopy groups of highly connected manifolds

Abstract

For n ≥ 2 we consider (n − 1)-connected closed manifolds of dimension at most (3n − 2). We prove that away from a finite set of primes, the p-local homotopy groups of M are determined by the dimension of the space of indecomposable elements in the cohomology ring H * (M ). Moreover, we show that these p-local homotopy groups can be expressed as direct sum of p-local homotopy groups of spheres. This generalizes some of the results of our earlier work [1].

The Homotopy Type of the Loops on $$(n-1)$$ -Connected $$(2n+1)$$ -Manifolds

“…There is an explicit expression for calculating the number of π k (S l ) (p) in π k (M ) (p) in Theorem 4.9. This expression is quite similar to [3,Theorem 3.6,Theorem 3.7] where the computation is carried out in the general case of (n − 1)-connected d-manifolds with d ≤ 3n − 2. A closer inspection shows that the results in Theorem 4.5 and Theorem 4.9 are stronger for (n − 1)-connected (2n + 1)-manifolds.…”

“…It follows that in the expression of Theorem 4.9 for arbitrarily large l, π * S l (p) occurs as a summand of π * M (p) . Now we observe [17] that any p s may occur as the order of an element in π * S l for arbitrarily large l. 3 In section 5, using a different method we verify that Ω(S n ∨ S n+1 ) is a retract of ΩM if r ≥ 2 (Corollary 5.5). It follows from [18] that for such a M , π * M has summands π * S k for k arbitrarily large, so they cannot have homotopy exponents at any prime p.…”

“…Under suitable torsion free assumptions (n−1)-connected (2n+1)-manifolds and even more generally (n−1)-connected d-manifolds with d ≤ 3n − 2 are also of this type. In recent times there have been a number computations for these manifolds [6,5,3] which we recall now.…”

“…If the Betti number is 1, n is forced to be 2, 4 or 8 by the Hopf invariant one problem [1], and in this case we have observed that an analogous result is true only after inverting finitely many primes. These arguments have also been carried out for (n − 1)-connected d-manifolds with d ≤ 3n − 2 after inverting finitely many primes [3].…”

“…A closer inspection shows that the results in Theorem 4.5 and Theorem 4.9 are stronger for (n − 1)-connected (2n + 1)-manifolds. For, in [3] the result about the number of primes being inverted in the expression above is not determined from its homology, while in the current paper we need to invert only those primes which appear as orders of elements in the torsion subgroup G.…”