Free loop space homology of highly connected manifolds

Alexander Berglund; Kaj Börjeson

doi:10.1515/forum-2015-0074

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Free loop space homology of highly connected manifolds

Alexander Berglund and Kaj Börjeson

Published/Copyright: June 14, 2016

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From the journal Forum Mathematicum Volume 29 Issue 1

Abstract

We calculate the homology of the free loop space of (n-1)-connected closed manifolds of dimension at most 3⁢n-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 [6].

Keywords: Free loop spaces; Koszul algebras; string topology; BV-algebras; Hochschild cohomology

MSC 2010: 55P50; 16S37; 55P35; 16E40

Communicated by Frederick R. Cohen

A Homological perturbation theory

Definition A.1

Suppose (C,dC) and (D,dD) are chain complexes with maps f:C→D, g:D→C, and a map h:C→C with |f|=|g|=0 and |h|=1,

Suppose moreover that f and g are chain maps and that

dC⁢h+h⁢dC=g⁢f-idC,f⁢g=idD,

and

f⁢h=0,h⁢h=0,h⁢g=0.

Call a diagram like this a contraction with data (C,D,dC,dD,f,g,h).

Remark A.2

It is harmless to assume the extra identities

f⁢h=0,h⁢h=0,h⁢g=0.

Suppose we have data satisfying all the above identities except these. In [28] it is noted that we can always redefine h such that these are satisfied.

Definition A.3

Given a complex with differential d, we say that a perturbation of d is a map t of degree -1 on the same complex such that (d+t)2=0.

Theorem A.4

Theorem A.4 (Basic perturbation lemma)

Suppose given a contraction as in Definition A.1 and a perturbation t of dC. If 1-h⁢t is invertible, then, setting Σ=t⁢(1-h⁢t)-1, there is a new contraction with data (C,D,dC+t,dD+t′,f′,g′,h′),

where

t′=f⁢Σ⁢g,f′=f+f⁢Σ⁢h,g′=g+h⁢Σ⁢g,h′=h+h⁢Σ⁢h.

Proof.

See [8], [21] and [2]. ∎

B A∞-structures and formality

Definition B.1

Let 𝕜 be a commutative ring. An A∞-algebra structure on a graded 𝕜-module V consists of a collection of maps mk:A⊗k→A of degree k-2 such that the following identity is satisfied for every k≥1:

∑(-1)r+s⁢t⁢mr+1+t⁢(id⊗r⊗ms⊗id⊗t)=0.

The sum is over all r,s,t such that r+s+t=k.

Remark B.2

Note that if mk=0 for all k≥3, this is equivalent to the data of a usual dg-algebra (with d=m1).

Theorem B.3

Theorem B.3 (Homotopy transfer theorem)

Given a contraction

where A is a dg-algebra with multiplication m, there is an A∞-structure {mk}k≥1 on the homology H*⁢(A), where mk is given by an alternating sum over all rooted trees with k leaves as described in [27]. If mk=0 for every k≥3, then A is a formal dg-algebra.

Proof.

See [27] for a direct proof, or [5] for a proof using homological perturbation theory. That A is formal if the higher products vanish follows from [25]. ∎

Example B.4

The map m3 is described by the following expression:

The formulas for m4,m5,… are similar.

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Received: 2015-4-22

Revised: 2016-3-21

Published Online: 2016-6-14

Published in Print: 2017-1-1

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https://doi.org/10.1515/forum-2015-0074

Keywords for this article

Free loop spaces; Koszul algebras; string topology; BV-algebras; Hochschild cohomology