Equivariant maps related to the topological Tverberg conjecture.

*(English)*Zbl 1382.55009The well-known topological Tverberg conjecture states that for every map \(f\) from the simplex \(\Delta^{(d+1)(n-1)}\) to \(\mathbb{R}^d\), there are \(n\) disjoint faces whose images have a common intersection point. The first proof of the conjecture for \(n\) a prime power was given by M. Özaydin [“Equivariant maps for the symmetric group”, Preprint, http://digital.library.wisc.edu/1793/63829].

Let \(N = (d+1)(n-1)\) and \(\Sigma_n\) the permutation group on \(n\) elements. A map violating the topological Tverberg conjecture leads to a \(\Sigma_n\)-equivariant map from the \(n\)-fold deleted join of \(\Delta^N\) to the complement of a diagonal subspace in the \(n\)-fold join of \(\mathbb{R}^d\). After making certain identifications one obtains a \(\Sigma_n\)-equivariant map from the \((N + 1)\)-fold join \(\{1,\dots, n\}^{*(N+1)}\) to \(S(W^d)\), where \(W\) denotes the standard representation of \(\Sigma_n\) and \(S(W^d)\) its representation sphere. In the same work, Özaydin observed that when \(n\) is not a prime, the \(C_n\)-equivariant maps \(\{1,\dots, n\}^{*(N+1)}\) to \(S(\bar{\rho}^d)\) do exist for \(N = (d + 1)(n -1)\), where \(\bar{\rho}\) is the restriction to \(C_n\) of the standard representation \(W\) of \(\Sigma_n\).

In view of the recent counterexamples to the topological Tverberg conjecture when \(n\) is not a prime power, it is natural to ask whether a weaker version of the conjecture holds in this case. More precisely, Özaydin asked the following question: For a subgroup \(G < \Sigma_n\), does there exist a \(G\)-equivariant map \(\{1,\dots, n\}^{*(N+1)}\) to \(S(W^d)\) with \(N > (d + 1)(n - 1)\)?

The aim of the paper under review is to investigate the preceding question for cyclic groups, products of elementary abelian groups and dihedral groups. The authors prove that such equivariant maps exist in these cases. As applications, they also investigate Borsuk-Ulam properties for cyclic and dihedral groups.

Let \(N = (d+1)(n-1)\) and \(\Sigma_n\) the permutation group on \(n\) elements. A map violating the topological Tverberg conjecture leads to a \(\Sigma_n\)-equivariant map from the \(n\)-fold deleted join of \(\Delta^N\) to the complement of a diagonal subspace in the \(n\)-fold join of \(\mathbb{R}^d\). After making certain identifications one obtains a \(\Sigma_n\)-equivariant map from the \((N + 1)\)-fold join \(\{1,\dots, n\}^{*(N+1)}\) to \(S(W^d)\), where \(W\) denotes the standard representation of \(\Sigma_n\) and \(S(W^d)\) its representation sphere. In the same work, Özaydin observed that when \(n\) is not a prime, the \(C_n\)-equivariant maps \(\{1,\dots, n\}^{*(N+1)}\) to \(S(\bar{\rho}^d)\) do exist for \(N = (d + 1)(n -1)\), where \(\bar{\rho}\) is the restriction to \(C_n\) of the standard representation \(W\) of \(\Sigma_n\).

In view of the recent counterexamples to the topological Tverberg conjecture when \(n\) is not a prime power, it is natural to ask whether a weaker version of the conjecture holds in this case. More precisely, Özaydin asked the following question: For a subgroup \(G < \Sigma_n\), does there exist a \(G\)-equivariant map \(\{1,\dots, n\}^{*(N+1)}\) to \(S(W^d)\) with \(N > (d + 1)(n - 1)\)?

The aim of the paper under review is to investigate the preceding question for cyclic groups, products of elementary abelian groups and dihedral groups. The authors prove that such equivariant maps exist in these cases. As applications, they also investigate Borsuk-Ulam properties for cyclic and dihedral groups.

Reviewer: Mahender Singh (Manauli)