Free loop space homology of highly connected manifolds

Abstract

Abstract. We calculate the homology of the free loop space of (n − 1)-connected closed manifolds of dimension at most 3n − 2 (n ≥ 2), with the Chas-Sullivan loop product and loop bracket. Over a field of characteristic zero, we obtain an expression for the BV-operator. We also give explicit formulas for the Betti numbers, showing they grow exponentially. Our main tool is the connection between formality, coformality and Koszul algebras that was elucidated by the first author [Ber14a].

Non-Formality of Planar Configuration Spaces in Characteristic 2

“…Definition 6.2. The convolution algebra structure on Hom(W, H) uses the coproduct structure on the domain and the product on the range to obtain an associative product of degree 1 [4]. For f, g : W → H, we have the composition Proof.…”

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

“…map onto the generators on H n (Q) and H n+1 (Q) respectively. We easily compute as in [8] H * (ΩQ) ∼ = Z[u, v] with |u| = n − 1 and |v| = n so that the map Ωλ : ΩS n → ΩQ sends the generator in H n−1 (ΩS n ) to u, and the map Ωλ ′ : ΩS n+1 → ΩQ sends the generator in H n (ΩS n+1 ) to v. The composite…”

“…M (t) is the generating series for ΩM [22, Theorem 3.5.1], from the fact that H * (ΩM ; D M ) is Koszul as an associative algebra[8]. Let η m := coefficient of t m in log(q M (t)).…”

The free loop space homology of $$(n-1)$$ ( n - 1 ) -connected 2n-manifolds

“…Berglund and Borjeson [6] have subsequently computed the free loop space homology of highly connected manifolds (including the ones considered in this paper) using different techniques. They also give a description of the action of the BV-operator and the Chas-Sullivan loop product.…”