[Paper Review] Hodge decomposition in the rational homology and homotopy of high dimensional string links

Paul Arnaud Songhafouo Tsopméné, Victor Turchin|arXiv (Cornell University)|Apr 3, 2015

Homotopy and Cohomology in Algebraic Topology22 references4 citations

TL;DR

This paper extends Hodge decomposition results from high-dimensional long knots to high-dimensional string links, establishing rational homology and homotopy splittings via finite graph-complexes. It computes the generating functions for Euler characteristics of homological and homotopical summands, and determines the supertrace of the symmetric group action on positive arity components of the modular envelope of the L∞ operad.

ABSTRACT

Arone and Turchin obtained the Hodge decomposition in the homology and homotopy of the high dimensional anologues of spaces of long knots. Next they showed that when the dimensions are in the stable range, the rational homology and homotopy of these latter spaces can be calculated as the homology of a direct sum of certain finite graph-complexes that they described explicitly. Finally they computed the generating function of the Euler characteristics of the summands in the homological splitting. In this paper, we generalize all these results to high dimensional analogues of spaces of string links. We also provide the generating function of the Euler characteristics of the summands in the homotopical splitting. As a byproduct result of these computations we also determine the supertrace of the symmetric group action on the positive arity components of the modular envelop of the L(infinity) operad.

Motivation & Objective

• Generalize Hodge decomposition results from high-dimensional long knots to high-dimensional string links.
• Establish rational homology and homotopy splittings using finite graph-complexes in the stable range.
• Compute the generating function for the Euler characteristics of the homological summands.
• Provide the generating function for the Euler characteristics of the homotopical summands.
• Determine the supertrace of the symmetric group action on positive arity components of the modular envelope of the L∞ operad.

Proposed method

• Adapt the Hodge decomposition framework from long knots to string links using operadic and homotopical tools.
• Construct explicit finite graph-complexes that model the rational homology and homotopy of high-dimensional string link spaces.
• Apply stable homotopy theory to ensure the splitting holds in the stable dimension range.
• Use generating functions to encode the Euler characteristics of the summands in both homological and homotopical splittings.
• Leverage operadic duality and modular envelopes to analyze the symmetric group action on L∞ operad components.
• Compute the supertrace via representation-theoretic techniques on the positive arity parts of the modular envelope.

Experimental results

Research questions

• RQ1 How can the Hodge decomposition framework be extended from long knots to high-dimensional string links?
• RQ2 What finite graph-complexes model the rational homology and homotopy of high-dimensional string link spaces?
• RQ3 What is the generating function for the Euler characteristics of the homological summands in the splitting?
• RQ4 What is the generating function for the Euler characteristics of the homotopical summands in the splitting?
• RQ5 What is the supertrace of the symmetric group action on the positive arity components of the modular envelope of the L∞ operad?

Key findings

• The rational homology and homotopy of high-dimensional string link spaces decompose into direct sums of finite graph-complexes in the stable range.
• The generating function for the Euler characteristics of the homological summands is explicitly computed.
• The generating function for the Euler characteristics of the homotopical summands is derived.
• The supertrace of the symmetric group action on the positive arity components of the modular envelope of the L∞ operad is determined.
• The results generalize prior work by Arone and Turchin from long knots to string links, extending the scope of Hodge-theoretic decompositions in higher dimensions.