[Paper Review] On Brunnian-type links and the link invariants given by homotopy groups of spheres

Jie Wu|ArXiv.org|Sep 27, 2009

Homotopy and Cohomology in Algebraic Topology26 references4 citations

TL;DR

This paper introduces a novel link invariant using the quotient of intersection subgroups and symmetric commutator subgroups in link groups, showing that this quotient is isomorphic to homotopy groups of spheres for certain Brunnian-type links. The construction yields invariants that detect links beyond Milnor's invariants and realizes all homotopy groups of spheres via geometric Massey products on links.

ABSTRACT

We introduce the (general) homotopy groups of spheres as link invariants for Brunnian-type links through the investigations on the intersection subgroup of the normal closures of the meridians of strongly nonsplittable links. The homotopy groups measure the difference between the intersection subgroup and symmetric commutator subgroup of the normal closures of the meridians and give the invariants of the links obtained in this way. Moreover the higher homotopy-group invariants can produce some links that could not be detected by the Milnor invariants. Furthermore all homotopy groups of spheres can be obtained from the geometric Massey products on links.

Motivation & Objective

• To define new link invariants using the quotient of intersection subgroups and symmetric commutator subgroups in link groups.
• To establish a connection between link invariants and homotopy groups of spheres through geometric constructions.
• To demonstrate that higher homotopy group invariants can detect links not detectable by Milnor invariants.
• To show that all homotopy groups of spheres arise from geometric Massey products on links.
• To provide a homotopy-theoretic interpretation of the link invariant via colimits of n-corners and mapping cones.

Proposed method

• Define the intersection subgroup $ A(L, L') $ as the intersection of normal closures of meridians of components in a sublink $ L' $ of a link $ L $.
• Define the symmetric commutator subgroup $ A_S[L, L'] $ as the group generated by $ t $-fold iterated commutators of elements from the normal closures of meridians.
• Form the quotient group $ \mathcal{A}(L, L') = A(L, L') / A_S[L, L'] $, which serves as the primary link invariant.
• Use the theory of $ n $-corners and homotopy colimits to construct a map $ \theta(f) \colon S^n \to S^3 \setminus L_0 $ from loops in the link complement.
• Establish an isomorphism $ \mathcal{A}(L, L) \cong \pi_n(S^3) $ when $ L_0 = \emptyset $ and $ L' = L $, and a more complex isomorphism involving wedge sums of spheres and smash products when $ L_0 \neq \emptyset $.
• Use the naturality of the homotopy colimit construction and the connectivity of $ n $-corners to show that the induced map $ \theta(f) $ represents the invariant via the map $ \rho_{\mathbf{M}(L)} $.

Experimental results

Research questions

• RQ1 Can the quotient $ \mathcal{A}(L, L') $ provide new invariants for Brunnian-type links beyond Milnor invariants?
• RQ2 How are homotopy groups of spheres realized as link invariants through geometric constructions in link complements?
• RQ3 What is the role of strongly nonsplittable link pairs in realizing nontrivial homotopy invariants?
• RQ4 Can all homotopy groups of spheres be obtained via geometric Massey products on links?
• RQ5 What is the precise algebraic-topological structure of $ \mathcal{A}(L, L') $ for different configurations of $ L $ and $ L' $?

Key findings

• For a strongly nonsplittable pair $ (L, L_0) $ with $ L_0 = \emptyset $ and $ L' = L $, the invariant $ \mathcal{A}(L, L) $ is isomorphic to $ \pi_n(S^3) $, providing a direct link between link invariants and homotopy groups of spheres.
• When $ L_0 \neq \emptyset $ and is splittable with splitting genus $ \nu \geq 1 $, $ \mathcal{A}(L, L \setminus L_0) \cong \pi_n\left( \bigvee_{j=1}^\nu G(L_0) \ltimes S^2 \right) $, which decomposes into a sum of homotopy groups of wedge sums of spheres and iterated smash products.
• The construction of $ \theta(f) $ via homotopy colimits of $ n $-corners yields a well-defined map $ S^n \to S^3 \setminus L_0 $ whose homotopy class corresponds to the invariant $ \rho_{\mathbf{M}(L)}(\alpha) $, linking the geometric construction to the algebraic invariant.
• The invariant $ \mathcal{A}(L, L') $ vanishes unless $ L_0 = \emptyset $ and $ L' = L $, or $ L' = L \setminus L_0 $ with $ L_0 $ nonempty and splittable, showing a precise structural constraint.
• All elements of $ \pi_*(S^3) $ can be realized as invariants of $ (n+1) $-links via this construction, demonstrating that the method realizes the full range of homotopy groups of spheres.
• The geometric Massey product construction via $ \theta(f) $ generalizes classical Massey products and provides a geometric realization of higher homotopy groups in link theory.