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)