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

Journal of Homotopy and Related Structures, May 2016

Our goal in this paper is to compute the integral free loop space homology of \((n-1)\)-connected 2n-manifolds. We do this when \(n\ge 2\) and \(n\ne 2,4,8\), 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.

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

The free loop space homology of (n − 1)-connected 2n-manifolds Piotr Beben 0 1 Nora Seeliger 0 1 Mathematics Subject Classification 0 1 B Piotr Beben 0 1 0 Max-Planck-Institut für Mathematik , Vivatsgasse 7, 53111 Bonn , Germany 1 School of Mathematics, University of Southampton , Southampton SO17 1BJ , UK 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. String topology · Free loop space · Highly connected manifolds Let L X = map(S1, X ) denote the free loop space on X . This space comes equipped with an action ν : S1 × L X −→ L X that rotates loops, and an induced degree 1 homomorphism Communicated by Pascal Lambrechts. 1 Introduction : H∗(L X ) −→ H∗+1(L X ) known as the BV-operator, defined by setting and Sullivan [ 9 ] constructed a pairing (a) = ν∗([S1] ⊗ a). In addition Chas Hp(LX ) ⊗ Hq (LX ) −→ Hp+q−d (LX ) on a closed oriented d-manifold X that (together with the BV-operator) turns the shifted homology H∗(LX ) = H∗+d (LX ) into a Batalin–Vilkovisky (BV)-algebra. Batalin–Vilkovisky algebras have been computed in only a few special cases. One of the more general results to date (due to Felix and Thomas [ 12 ]) states that over a field F of characteristic zero and 1-connected X , H∗(LX ; F ) is isomorphic to a BV-algebra structure defined on the Hochschild cohomology H H ∗(C ∗(X ), C ∗(X )). Unfortunately, this theorem is generally not true for fields with nonzero characteristic [ 20 ]. Beyond these results, the BV-algebra over various coefficient rings has been completely determined for spheres [ 10,20,25 ], certain Stiefel manifolds [ 24 ], Lie groups [ 17 ], and projective spaces [ 10,16,22,27,28 ], using a mixture of techniques ranging from homotopy theoretic to geometric, as well as the well-known connections to Hochschild cohomology. In this paper we focus on the free loop space homology of highly connected 2nmanifolds, together with the action of the BV-operator. The coefficient ring R for homology and cohomology is assumed to be either any field, or the integers Z, but we suppress it from notation most of the time. Fix n ≥ 2, M a (n − 1)-connected, closed, oriented 2n-manifold with H n(M ) of rank m ≥ 1. Let C = [ci j = ai ∪ a j , [M ] ] be the m × m matrix for the intersection form H n(M ) × H n(M ) −→ Z with respect to some choice of basis {a1, . . . , am } for H n(M ) (we use the same notation for the dual basis of H n(M )). This form is nonsingular, symmetric when n is even, and skew-symmetric when n is odd. Denote H n(M ) and H 2n(M ) ∼= Z by the free graded modules R-modules A = R{a1, . . . , am } and K = R{[M ]}, and the desuspension of A by V = R{u1, . . . , um } with |ui | = n − 1. Let T (V ) = R ⊕ i≥1 V ⊗i be the free tensor algebra generated by V , and I be the two-sided ideal of the tensor algebra T (V ) generated by the following degree 2n − 2 element χ = where [x , y] = x y − (−1)|x||y| yx denotes the graded Lie bracket in T (V ). Take the quotient algebra U = and the degree −1 maps of graded R-modules d : A ⊗ U −→ U and d : K ⊗ U −→ A ⊗ U , which are given for any y ∈ U by the formulas d(ai ⊗ y) = [ui , y] d ([M ] ⊗ y) = One can think of W by first taking the R-submodule W of −1 A ⊗ T (V ) ∼= T (V ) generated by elements that are invariant modulo I under graded cyclic permutations, that is, invariant after projecting to U . Then W is the projection of W onto ( A ⊗ U )/Im d . Our main result is that the homology of this chain complex is the integral free loop space homology of M under some conditions: Theorem 1.1 Suppose n ≥ 2, n = 2, 4, 8, and m ≥ 1. Then there exists an isomorphism of graded R-modules H∗(LM ) ∼= Q ⊕ W ⊕ Z. The restriction away from 2, 4, and 8 traces back to an argument that we use to determine H∗( M ), which does not apply to situation where there are cup product squares equal to the fundamental class [M ], or −[M ]. Failure of a degree placement argument to compute certain differentials is another reason that we restrict away from n = 2. We also determine the action of the BV-operator on H∗(LM ; Q), in a sense, up-toabelianization of U when n > 3 is odd. η Consider the graded abelianization map T (V ) −→ S(V ), where S(V ) is the free η graded symmetric algebra generated by V . Since η(χ ) = 0, η factors through U −→ S(V ). Also, consider the maps A ⊗ U 1−A→⊗η A ⊗ S(V ) and K ⊗ U 1−K→⊗η K ⊗ S(V ). Since (1A ⊗η)◦d = 0 and η◦d = 0, then η and these two maps induce abelianization maps ηq Q −→ S(V ), W −η→w A ⊗ S(V ), ηz Z −→ K ⊗ S(V ). Theorem 1.2 Let n > 3 be odd. The BV operator : H∗(LM ; Q) −→ H∗+1(LM ; Q) satisfies (Q) ⊆ W and (W) ⊆ Z, and (Z) = {0}. Moreover, the composite ηw Q −→ W −→ A ⊗ S(V ) is given by k j=1 k j=1 ηw ◦ (1 ⊗ (ui1 . . . uik )) = ai j ⊗ (ui1 (...truncated)


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Piotr Beben, Nora Seeliger. The free loop space homology of \((n-1)\) -connected 2n-manifolds, Journal of Homotopy and Related Structures, 2017, pp. 413-432, Volume 12, Issue 2, DOI: 10.1007/s40062-016-0132-4