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

Abstract

Our goal in this paper is to compute the integral free loop space homology of (n −1)-connected 2n-manifolds. We do this when n ≥ 2 and n = 2, 4, 8, though the techniques here should cover a much wider range of manifolds. We also give partial information concerning the action of the Batalin-Vilkovisky operator.

The cohomology of the free loop spaces of $SU(n+1)/T^n$

“…However such computations will be unnecessary in the present work. During a Gröbner basis computation of the intersection of ideals in part (2), expression containing elements in E * ,b r for b > a may be considered, these can be discarded immediately, since reduction, S-polynomials and G-polynomials of such elements can never decease the row b during the procedure without reducing the entire polynomial to zero. Doing this greatly speeds up the execution of the algorithm.…”

“…In studying string topology operations the homology of the free loop space has also been considered in a several additional cases. Integral homology of the free loop space of complex Stiefel manifold [38], spaces whose cohomology is an exterior algebra with field coefficients [6], (n − 1)-connected manifolds up to dimension 3n − 2 with homology coefficients over a field [4] and many cases of (n − 1)-connected 2n-manifolds with integral coefficients [2]. Cohen-Jones-Yan [15] have also shown that there is a spectral sequence of algebras converging to the homology of the free space and the loop product demonstrating its use on spheres and complex proactive spaces.…”

“…We begin by proving part(2) in the case l = 1, then extend to the general case. Components of proof of part (2) are then used to prove part (1) in the case l = 1.…”

“…1 −! (x 6 ) b )s 6 , (x 2 ) a2 (x 4 ) a4 (x 6 ) a6 (γ 2 1 + γ1 γ2 + γ2 2 + 4(γ 1 + γ2 )γ + 6γ 2 ), (x 2 ) a2 (x 4 ) a4 (x 6 ) a6 (γ 3 1 + 4γ 2 1 γ + 6γ 1 γ2 + 4γ 3 ), (x 2 ) a2 (x 4 ) a4 (x 6 ) a6 γ4 , (x 2 ) a2 (x 4 ) a4 (x 6 ) a6 (y 1 γ1 + y 2 γ2 + y 3 (4γ − γ1 − γ2 )), (x 2 ) a2 (x 4 ) a4 (x 6 ) a6 (y…”

Free loop space homology of highly connected manifolds

“…Theorem 1.3 generalizes [BB13, Theorem C(1)]. Free loop space homology of (n − 1)-connected 2n-dimensional manifolds has been studied in [BS12] using different methods, but the calculations there are not complete.…”