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
S
1
;Q) → HH∗
(∧V;C) is injective, where X
S
1
is the space of free loops on
X, and H∗(X
S
1
;Q) is the loop space homology.
Oteng, M (2024). Loop space homology of elliptic spaces. Afribary. Retrieved from https://afribary.com/works/loop-space-homology-of-elliptic-spaces
Oteng, Maphane "Loop space homology of elliptic spaces" Afribary. Afribary, 30 Mar. 2024, https://afribary.com/works/loop-space-homology-of-elliptic-spaces. Accessed 24 Nov. 2024.
Oteng, Maphane . "Loop space homology of elliptic spaces". Afribary, Afribary, 30 Mar. 2024. Web. 24 Nov. 2024. < https://afribary.com/works/loop-space-homology-of-elliptic-spaces >.
Oteng, Maphane . "Loop space homology of elliptic spaces" Afribary (2024). Accessed November 24, 2024. https://afribary.com/works/loop-space-homology-of-elliptic-spaces