[Paper Review] The Borsuk-Ulam property for homotopy classes of selfmaps of surfaces of Euler characteristic zero

Daciberg Lima Gonçalves, John Guaschi+1 more|arXiv (Cornell University)|Aug 1, 2016

Homotopy and Cohomology in Algebraic Topology19 references13 citations

TL;DR

This paper investigates the Borsuk-Ulam property for homotopy classes of self-maps on surfaces of Euler characteristic zero—specifically the 2-torus (T²) and Klein bottle (K²)—under free involutions. It uses braid group theory and fundamental group homomorphisms to classify which homotopy classes satisfy the property: for T² with orientation-reversing involutions, the property holds iff the induced matrix has (β₁,₁, β₂,₁) ≠ (0,0) and β₁,₂, β₂,₂ even; for K², a class has the property iff it lifts to the torus. The results provide a complete homotopy classification for these surfaces under free involutions.

ABSTRACT

Let M and N be topological spaces such that M admits a free involution $$ au $$ τ . A homotopy class $$\beta \in [ M , N ] $$ β ∈ [ M , N ] is said to have the Borsuk–Ulam property with respect to $$ au $$ τ if for every representative map $$f:\,M ightarrow N$$ f : M → N of $$\beta $$ β , there exists a point $$x \in M$$ x ∈ M such that $$f ( au ( x) ) = f(x)$$ f ( τ ( x ) ) = f ( x ) . In the case where M is a compact, connected manifold without boundary and N is a compact, connected surface without boundary different from the 2-sphere and the real projective plane, we formulate this property in terms of the pure and full 2-string braid groups of N, and of the fundamental groups of M and the orbit space of M with respect to the action of $$ au $$ τ . If $$M=N$$ M = N is either the 2-torus $$\mathbb {T}^2$$ T 2 or the Klein bottle $$\mathbb {K}^2$$ K 2 , we then solve the problem of deciding which homotopy classes of [M, M] have the Borsuk–Ulam property. First, if $$ au :\,\mathbb {T}^2 ightarrow \mathbb {T}^2$$ τ : T 2 → T 2 is a free involution that preserves orientation, we show that no homotopy class of $$[ \mathbb {T}^2, \mathbb {T}^2]$$ [ T 2 , T 2 ] has the Borsuk–Ulam property with respect to $$ au $$ τ . Second, we prove that up to a certain equivalence relation, there is only one class of free involutions $$ au :\,\mathbb {T}^2 ightarrow \mathbb {T}^2$$ τ : T 2 → T 2 that reverse orientation, and for such involutions, we classify the homotopy classes in $$[\mathbb {T}^2, \mathbb {T}^2]$$ [ T 2 , T 2 ] that have the Borsuk–Ulam property with respect to $$ au $$ τ in terms of the induced homomorphism on the fundamental group. Finally, we show that if $$ au :\,\mathbb {K}^2 ightarrow \mathbb {K}^2$$ τ : K 2 → K 2 is a free involution, then a homotopy class of $$[\mathbb {K}^2, \mathbb {K}^2]$$ [ K 2 , K 2 ] has the Borsuk–Ulam property with respect to $$ au $$ τ if and only if the given homotopy class lifts to the torus.

Motivation & Objective

• Understand which homotopy classes of self-maps on surfaces with Euler characteristic zero satisfy the Borsuk-Ulam property under free involutions.
• Address the refinement of the Borsuk-Ulam problem: when does a homotopy class β ∈[M, N] satisfy f(τ(x)) = f(x) for some x, for every representative f of β?
• Provide a complete classification of homotopy classes with the Borsuk-Ulam property for M = N = T² or K² and free involutions τ.
• Establish connections between the Borsuk-Ulam property and lifting properties to the universal cover (the torus) for the Klein bottle.
• Use algebraic topology tools—braid groups, fundamental groups, and homomorphisms—to characterize the property in terms of group-theoretic conditions.

Proposed method

• Formulate the Borsuk-Ulam property for homotopy classes using the pure and full 2-string braid groups of the target surface N.
• Use the fundamental group π₁(M) and the orbit space M/τ to relate the Borsuk-Ulam condition to group homomorphisms and exact sequences.
• Analyze the induced homomorphism β# on π₁(T²) ≅ ℤ ⋊ ℤ, represented by an integral matrix, to determine the Borsuk-Ulam property for orientation-preserving and orientation-reversing involutions.
• Apply covering space theory and the lifting criterion: for K², a map β has the Borsuk-Ulam property iff it lifts to the torus via the double cover T² → K².
• Use short exact sequences involving braid groups B₂(N) and P₂(N), and the action of the symmetric group S₂, to model the homotopy lifting and orbit structure.
• Employ group-theoretic conditions on elements of P₂(K²) and B₂(K²) to verify the existence of maps satisfying the Borsuk-Ulam condition.

Experimental results

Research questions

• RQ1 Which homotopy classes β ∈[T², T²] have the Borsuk-Ulam property with respect to a free orientation-preserving involution τ?
• RQ2 Which homotopy classes β ∈[T², T²] have the Borsuk-Ulam property with respect to a free orientation-reversing involution τ, and how can this be characterized algebraically?
• RQ3 Does the Borsuk-Ulam property for self-maps of the Klein bottle K² depend on whether the map lifts to the torus?
• RQ4 How do equivalence classes of free involutions on T² affect the classification of homotopy classes with the Borsuk-Ulam property?
• RQ5 What is the relationship between the Borsuk-Ulam property and the induced homomorphism on fundamental groups for surfaces of Euler characteristic zero?

Key findings

• For the 2-torus T² with a free orientation-preserving involution τ, no homotopy class β ∈[T², T²] has the Borsuk-Ulam property.
• For a free orientation-reversing involution τ₂ on T², a homotopy class β has the Borsuk-Ulam property if and only if the induced matrix (βᵢⱼ) satisfies (β₁,₁, β₂,₁) ≠ (0, 0) and β₁,₂, β₂,₂ are both even.
• Up to equivalence, there is only one class of free orientation-reversing involutions on T², so the classification applies uniformly to all such involutions.
• For the Klein bottle K², a homotopy class β ∈[K², K²] has the Borsuk-Ulam property with respect to a free involution τ if and only if β lifts to the torus via the double cover T² → K².
• The lifting condition is equivalent to the induced homomorphism β# : ℤ ⋊ ℤ → ℤ ⋊ ℤ being of Type B in the classification of group homomorphisms.
• Homeomorphisms of K² preserve the Borsuk-Ulam property: if f₁ lifts to the torus, then f₂ = f₁ ◦ H⁻¹ also lifts, so the property is invariant under homeomorphism.