[Paper Review] Gauss-type formulas for link map invariants

Sergey A. Melikhov|arXiv (Cornell University)|Nov 9, 2017

Homotopy and Cohomology in Algebraic Topology2 references8 citations

TL;DR

This paper establishes Gauss-type formulas for computing Koschorke's β-invariant and the triple μ-invariant of link maps in the critical dimension as degrees of specific maps between configuration spaces, analogous to the linking number. The key contribution is a geometric realization of these invariants via new operations akin to the Jin suspension, resolving long-standing expectations about their indirect computation through homotopy theory.

ABSTRACT

We find that Koschorke's $β$-invariant and the triple $μ$-invariant of link maps in the critical dimension can be computed as degrees of certain maps of configuration spaces - just like the linking number. Both formulas admit geometric interpretations in terms of Vassiliev's ornaments via new operations akin to the Jin suspension, and both were unexpected for the author, because the only known direct ways to extract $μ$ and $β$ from invariants of maps between configuration spaces involved some homotopy theory (Whitehead products and the stable Hopf invariant, respectively).

Motivation & Objective

• To provide explicit, geometric formulas for computing the β-invariant and triple μ-invariant of link maps in the critical dimension.
• To resolve the longstanding expectation that these invariants could only be extracted indirectly via homotopy-theoretic constructions such as Whitehead products and the stable Hopf invariant.
• To demonstrate that both invariants can be expressed as degrees of maps between configuration spaces, mirroring the classical Gauss linking number formula.
• To introduce new geometric operations, analogous to the Jin suspension, that clarify the topological meaning of these invariants.

Proposed method

• The paper defines a map φ_f from the product of configuration spaces to a sphere, using the link map f, and computes its equivariant degree to extract the β-invariant.
• It establishes that the β-invariant arises as the pullback of a generator of a twisted cohomology group, leading to a cohomological characterization.
• The method relies on the polynomial compactification of configuration spaces, which generalizes the Fulton–MacPherson construction and allows for the lifting of aligned maps.
• It uses the structure of the Fulton–MacPherson compactification and its stratification to define maps that preserve equivariance and stratum structure.
• The approach involves analyzing the behavior of maps under involution and identifying when the factor-exchanging involution is orientation-preserving, which determines the target cohomology group.
• The construction is validated through homotopy-theoretic consistency checks and by showing that embeddings that are link-homotopic yield trivial invariants.

Experimental results

Research questions

• RQ1 Can the β-invariant of a link map in the critical dimension be computed directly as a degree of a map between configuration spaces, without relying on homotopy-theoretic constructions?
• RQ2 Is there a geometric operation, analogous to the Jin suspension, that explains the structure of the β-invariant and μ-invariant in terms of configuration space maps?
• RQ3 Why were these invariants previously thought to require indirect homotopy-theoretic tools like Whitehead products and the stable Hopf invariant?
• RQ4 Do the configuration space integrals for μ and β admit a unified description similar to the Gauss linking number?
• RQ5 Can the polynomial compactification of configuration spaces serve as a natural framework for expressing higher-order link invariants?

Key findings

• The β-invariant of a link map in the critical dimension is equal to the equivariant degree of a specific map φ_f from the product of configuration spaces to a sphere, when k is odd.
• The triple μ-invariant can also be expressed as a degree of a map between configuration spaces, generalizing the Gauss linking number formula.
• For k odd, the β-invariant is non-zero only for k = 2, 4, 8, and the formula matches Koschorke’s original definition via double point counting.
• The paper constructs a geometric interpretation of the invariants using new operations akin to the Jin suspension, providing a direct topological meaning.
• The invariants vanish when one component of the link map is a PL embedding, as shown via homotopy factorization through the Hopf link.
• The method establishes that the invariants are well-defined and invariant under link homotopy, confirming their topological significance.