[Paper Review] The suspended free loop space of a symmetric space

Marcel Bökstedt, Iver Ottosen|arXiv (Cornell University)|Nov 3, 2005

Geometric Analysis and Curvature Flows12 references3 citations

TL;DR

This paper establishes a stable splitting of the suspension spectrum of the free loop space $Λ M_+$ for rank-one symmetric spaces $M = \mathbb{C}P^n$, $\mathbb{H}P^n$, and $\mathbb{O}P^2$, proving it is homotopy equivalent to the wedge sum of the suspension spectrum of $M_+$ and an infinite family of Thom spaces over the unit tangent sphere bundle of $M$. The result extends Ziller's homology splitting via Morse theory and equivariant homotopy theory, confirming a conjecture from prior work on mod 2 cohomology of loop spaces.

ABSTRACT

Let M be one of the projective spaces CP^n, HP^n for n>1 or the Cayley projective plane OP^2, and let LM denote the free loop space on M. Using Morse theory methods, we prove that the suspension spectrum of (LM)_+ is homotopy equivalent to the suspension spectrum of M_+ wedge a family of Thom spaces of explicit vector bundles over the tangent sphere bundle of M.

Motivation & Objective

• To prove a stable splitting of the suspension spectrum of the free loop space $\Lambda M_+$ for rank-one symmetric spaces.
• To confirm a conjecture from prior work that the mod 2 cohomology of $\Lambda M$ splits as a sum of Thom space cohomologies.
• To extend Ziller's homology splitting result to the level of spectra, achieving a homotopy equivalence rather than just a group-level isomorphism.
• To identify the explicit Thom spaces appearing in the splitting, particularly over the unit sphere bundle of the tangent bundle of $M$.
• To verify that the spectral splitting is compatible with Steenrod algebra actions and stable homotopy operations.

Proposed method

• Apply Morse theory to the free loop space $\Lambda M$ using the energy functional and the action of the isometry group.
• Use equivariant differential topology to identify subspaces of $\Lambda M$ as homogeneous spaces associated to symmetric space structure.
• Construct equivariant splittings of these homogeneous spaces by analyzing obstructions in representation ring cokernels.
• Show that the obstruction vanishes for $\mathbb{C}P^n$, $\mathbb{H}P^n$, and $\mathbb{O}P^2$, enabling stable splittings after suspension.
• Identify the summands in the splitting as Thom spaces of iterated Whitney sums of the pullback tangent bundle over the unit sphere bundle of $M$.
• Use cofiber sequences and homotopy equivalences involving Thom spaces to rewrite cohomological results from earlier work in spectral sequence terms.

Experimental results

Research questions

• RQ1 Does the suspension spectrum of the free loop space $\Lambda M_+$ for rank-one symmetric spaces admit a stable splitting?
• RQ2 Can the observed splitting in mod 2 cohomology of $\Lambda M$ be lifted to a splitting of spectra?
• RQ3 What are the explicit Thom spaces that appear as summands in the splitting of $\Sigma(\Lambda M_+)$?
• RQ4 Are the splittings compatible with the action of the Steenrod algebra and other cohomology operations?
• RQ5 What conditions on the obstruction in the representation ring must be satisfied for such a splitting to exist?

Key findings

• For $M = \mathbb{C}P^n$, there is a homotopy equivalence $\Sigma(\Lambda \mathbb{C}P^n)_+ \simeq \Sigma \mathbb{C}P^n_+ \vee \bigvee_{m=1}^\infty \Sigma \mathrm{Th}(\xi_m)$, where $\xi_m$ is a bundle over the unit sphere bundle of the tangent bundle.
• For $M = \mathbb{H}P^n$, the splitting involves Thom spaces of bundles involving the tangent bundle and a 3-dimensional bundle $\eta$ associated to the adjoint representation of $S^3$.
• For $M = \mathbb{O}P^2$, the splitting holds due to triviality of Stiefel-Whitney classes of the 7-dimensional bundle $\eta$ over the sphere bundle.
• The spectral splitting is compatible with the Steenrod algebra action, as confirmed by comparing with earlier cohomological results from [4].
• The obstruction to splitting lies in the cokernel of a map between representation rings and vanishes for the considered symmetric spaces.
• The result implies stable splittings for all generalized homology theories and provides a stronger statement than Ziller’s homology-level splitting.