Existence Of Equivariant Maps Research Articles

Stronger Counterexamples to the Topological Tverberg Conjecture

Denote by $$\Delta _M$$ the M-dimensional simplex. A map $$f:\Delta _M\rightarrow {{\mathbb {R}}}^d$$ is an almost r-embedding if $$f(\sigma _1)\cap \ldots \cap f(\sigma _r)=\emptyset $$ whenever $$\sigma _1,\ldots ,\sigma _r$$ are pairwise disjoint faces. A counterexample to the topological Tverberg conjecture asserts that if r is not a prime power and $$d\ge 2r+1$$ , then there is an almost r-embedding $$\Delta _{(d+1)(r-1)}\rightarrow {{\mathbb {R}}}^d$$ . This was improved by Blagojević–Frick–Ziegler using a simple construction of higher-dimensional counterexamples by taking k-fold join power of lower-dimensional ones. We improve this further (for d large compared to r): If r is not a prime power and $$N=(d+1)r-r\Big \lceil \dfrac{d+2}{r+1}\Big \rceil -2$$ , then there is an almost r-embedding $$\Delta _N\rightarrow {{\mathbb {R}}}^d$$ . The improvement follows from our stronger counterexamples to the r-fold van Kampen–Flores conjecture. Our proof is based on generalizations of the Mabillard–Wagner theorem on construction of almost r-embeddings from equivariant maps, and of the Özaydin theorem on existence of equivariant maps.

Vanishing of All Equivariant Obstructions and the Mapping Degree

Suppose that n is not a prime power and not twice a prime power. We prove that for any Hausdorff compactum X with a free action of the symmetric group $${\mathfrak {S}}_n$$ , there exists an $${\mathfrak {S}}_n$$ -equivariant map $$X \rightarrow {{\mathbb {R}}}^n$$ whose image avoids the diagonal $$\{(x,x,\dots ,x)\in {{\mathbb {R}}}^n\mid x\in {{\mathbb {R}}}\}$$ . Previously, the special cases of this statement for certain X were usually proved using the equivartiant obstruction theory. Such calculations are difficult and may become infeasible past the first (primary) obstruction. We take a different approach which allows us to prove the vanishing of all obstructions simultaneously. The essential step in the proof is classifying the possible degrees of $$\mathfrak S_n$$ -equivariant maps from the boundary $$\partial \Delta ^{n-1}$$ of $$(n-1)$$ -simplex to itself. Existence of equivariant maps between spaces is important for many questions arising from discrete mathematics and geometry, such as Kneser’s conjecture, the Square Peg conjecture, the Splitting Necklace problem, and the Topological Tverberg conjecture, etc. We demonstrate the utility of our result applying it to one such question, a specific instance of envy-free division problem.

Invariants in noncommutative dynamics

When a compact quantum group H coacts freely on unital C⁎-algebras A and B, the existence of equivariant maps A→B may often be ruled out due to the incompatibility of some invariant. We examine the limitations of using invariants, both concretely and abstractly, to resolve the noncommutative Borsuk–Ulam conjectures of Baum–Dąbrowski–Hajac. Among our results, we find that for certain finite-dimensional H, there can be no well-behaved invariant which solves the Type 1 conjecture for all free coactions of H. This claim is in stark contrast to the case when H is finite-dimensional and abelian. In the same vein, it is possible for all iterated joins of H to be cleft as comodules over the Hopf algebra associated to H. Finally, two commonly used invariants, the local-triviality dimension and the spectral count, may both change in a θ-deformation procedure.

The Borsuk-Ulam theorem for general spaces

Let X, Y be topological spaces and $T : X \rightarrow X$ a free involution. In this context, a question that naturally arises is whether or not all continuous maps $f : X \rightarrow Y$ have a T-coincidence point, that is, a point $x \in X$ with $f (x) = f (T (x))$. If additionally Y is equipped with a free involution $S : Y \rightarrow Y$ , another question is concerning the existence of equivariant maps $f : (X, T) \rightarrow (Y, S)$. In this paper we obtain results of this nature under cohomological (homological) conditions on the spaces X and Y.

On continuous maps between Grassmann manifolds

LetG n,k denote the Grassmann manifold ofk-planes in ℝn. We show that for any continuous mapf: G n,k→Gn,l the induced map inZ/2-cohomology is either zero in positive dimensions or has image in the subring generated by w1(γn, k), provided 1≤l<k≤[n/2] andn≥k+2l-1. Our main application is to obtain negative results on the existence of equivariant maps between oriented Grassmann manifolds. We also obtain positive results in many cases on the existence of equivariant maps between oriented Grassmann manifolds.