Borsuk-Ulam type theorems for manifolds (Q2845070)

The Borsuk-Ulam theorem states that any antipodal map \(f: S^n\to\mathbb{R}^n\) has at least one zero. This paper offers results of this type in a \(PL\) setting. If \(G\) is a finite group acting linearly on \(\mathbb{R}^n\) fixing only the origin and \(f: M^n\to\mathbb{R}^n\) is a equivariant transverse to zeros map on a closed, free \(G\) \(PL\)-manifold \(M^n\) with zero set \(Z_f\), then \(\deg_G(f):= 1\) if \(|Z_f|\) is an odd multiple of \(|G|\) and \(\deg_G(f):= 0\) if \(|Z_f|\) is an even multiple of \(|G|\).NEWLINENEWLINE The first major theorem asserts that if \(M^n\) is connected and \(h: N^n\to\mathbb{R}^n\) is an equivariant transverse to zeros map with \(N^n\) free \(G\)-cobordant to \(M^n\) and \(\deg_G(h)= 1\), then \(Z_f\) is not empty for any equivariant map \(f: M^n\to\mathbb{R}^n\). If \(M^n\) is a closed \(PL\)-manifold with free involution \(T\), then \(M^n\) is of Borsuk-Ulam type if any antipodal map \(f: M^n\to\mathbb{R}^n\) has at least one zero.NEWLINENEWLINE The second major result offers two conditions equivalent to Borsuk-Ulam type. The first is the existence of an antipodal transverse to zeros map \(h: M^n\to \mathbb{R}^n\) with \(\deg_{\mathbb{Z}_2}(h)= 1\) and the second is the condition that the coefficient of \([S^n, A]\) in the representation of the unoriented cobordism class \([M^n, T]\) is one. The classical Borsuk-Ulam theorem follows from the second condition.