Loop space homology of elliptic spaces

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Abstract:

In this thesis, we use the theory of minimal Sullivan models in rational homotopy theory to study the partial computation of the Lie bracket structure of the string homology on a formal elliptic space. In the process, we show the total space of the unit sphere tangent bundleS2m−1 → Ep→ Gk,n(C) over complex Grassmannian manifolds Gk,n(C) for 2 ≤ k ≤ n/2, where m = k(n − k) is not formal. This is done by exhibiting a non trivial Massey triple

product. On the other hand, let φ :(∧V,d) → (B,d) be a surjective morphism between com mutative differential graded algebras, where V is finite dimensional, and consider (B,d) a

module over ∧V via the mapping φ. We show that the Hochschild cohomology HH∗

(∧V;B)

can be computed in terms of the graded vector space of positive φ-derivations.

Given a Koszul Sullivan extension (∧V,d)

f ↣ (∧V ⊗ ∧W,d) = (C,d), we show that if

(∧V,d) is an elliptic 2-stage Postnikov tower Sullivan algebra, and if the natural homo morphism of the differential graded algebras (C,d) → (∧W,d¯) is surjective in homology,

then the natural graded linear map HH∗

(f) : HH∗

(∧V;∧V) → HH∗

(∧V;C), induced in

Hochschild cohomology by the inclusion (∧V,d)

f ↣ (C,d), is injective. In particular, if X

is an elliptic 2-stage Postnikov tower, and (∧V,d) is the minimal Sullivan model of X, then

HH∗

(f) : H∗(X

;Q) → HH∗

(∧V;C) is injective, where X

is the space of free loops on

X, and H∗(X

;Q) is the loop space homology.

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Loop space homology of elliptic spaces

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