Let \((X, \tau)\) be a free \(\mathbb Z_2\)-space, which means \(X\) is a topological space and \(\tau\) is a self-homeomorphism on \(X\) such that \(\tau^2\) is the identity map on \(X\). Such a map \(\tau\) is often refered to as an involution on \(X\).
Definition. Let \(X\), \(Y\) be topological spaces and \(\tau\) be an involution on \(X\). We call \((X, \tau, Y)\) a Borsuk-Ulam triple or say that it satisfies the Borsuk-Ulam property if and only if for any continuous map \(f: X \rightarrow Y\) there is \(x \in X\) such that \(f(x) = f(\tau(x))\).
In this paper, the authors study the Borsuk-Ulam theorem for triples \((X, \tau, \mathbb R^n)\) when \(X\) is a compact, connected, \(3\)-manifold and \(\tau\) is a fixed-point free involution.
Historically, the study of the theorem was initiated by a conjecture by St. Ulam; then it was proved by K. Borsuk in 1933 for the triple \((S^n, \tau, \mathbb R^n)\) (see [K. Borsuk, Fundam. Math. 20, 177–190 (1933; JFM 59.0560.01)]) when the involution \(\tau\) operates on the \(n\)-sphere \(S^n\). This study resulted in various generalizations and applications. For instance, if \(X = S^n\), then P. E. Conner and E. E. Floyd proved the theorem when \(Y\) is a finite smooth manifold in [Differentiable periodic maps. Berlin-Göttingen-Heidelberg: Springer-Verlag (1964; Zbl 0125.40103)]. Later, H. J. Munkholm [Ill. J. Math. 13, 116–124 (1969; Zbl 0164.53801)] showed that the assumption of smoothness can be replaced by assuming the manifold to be compact. Further, H. J. Munkholm [Math. Scand. 24, 167–185 (1969; Zbl 0186.57501)] and M. Nakaoka [Osaka J. Math. 7, 423–441 (1970; Zbl 0218.57011)] respectively have replaced \(S^n\) by closed topological \((\text{mod }2)\) \(n\)-homological spheres and \((\text{mod }p)\) \(n\)-homological spheres for \(X\).
The value of \(n\) for which the Borsuk-Ulam theorem holds in \((X, \tau, \mathbb R^n)\) is called the \(\mathbb Z_2\)-index. Roughly speaking, it is the value defined by \(\hbox{ind}_{\mathbb Z_2}(X, \tau)=\min\{n \in \mathbb N \cup \{\infty \} \mid\exists f: X \rightarrow S^n \}\), where \((X, \tau)\) is a free \(\mathbb Z_2\)-space.
A main goal of this article is to fully discuss the Borsuk-Ulam index according to cohomological operations applied to the characteristic class \(x \in H^1(N, \mathbb Z_2)\) where \(N = X/\tau\) is the orbit space or the quotient space. Note that if \(X\) is a compact connected \(3\)-manifold, then it is known for certain families of \(3\)-manifolds. For example, it is known for double covers of Seifert manifolds (see [A. Bauval et al., Zb. Pr. Inst. Mat. NAN Ukr. 6, No. 6, 165–189 (2013; Zbl 1313.55001)]), spherical manifolds (see [D. L. Gonçalves et al., J. Fixed Point Theory Appl. 9, No. 2, 285–294 (2011; Zbl 1323.55004)]). D. L. Gonçalves et al. [in: Group actions and homogeneous spaces. Proceedings of the international conference, Bratislava topology symposium “Group actions and homogeneous spaces”, Bratislava, Slovakia, September 7–11, 2009. Bratislava: Univ. Komenského, Fakulta Matematiky, Fyziky a Informatiky. 9–28 (2010; Zbl 1220.55001)] showed general cohomological conditions.
The authors review these general conditions in terms of an easily computable criterion with numerous examples, and many interesting results are contained in this article. The results below are quite interesting characterizations as for a family of \(L(1, 0)\) and \(L(0, 1)\), which are also denoted by \(S^3\) and \(S^1 \times S^2\), respectively.
Theorem 1. Let \(L(p,q)\) be a lens space such that \(p\) is even and the free \(\mathbb Z_2\)-space \((X, \tau)=(L(\frac{p}{2}, q), \tau)\) be as above. Further, let \(x \in H^1(L(p,q), \mathbb Z_2)\) be the non-trivial class. Then,
(1) \(\hbox{ind}_{\mathbb Z_2}(X, \tau)\geq 2\) and \((M, \tau, \mathbb R^2)\) is always a Borsuk-Ulam triple.
(2) \(\hbox{ind}_{\mathbb Z_2}(X, \tau)=3\) if and only if \(p \equiv 2 \bmod 4\).
Theorem 2. Let \(\tau\) be a free \(\mathbb Z_2\)-action on \(S^1 \times S^2\), and \(N\) be the quotient space of this action. Then,
(1) If \(N\) is \(S^1 \times S^2\), then \(\hbox{ind}_{\mathbb Z_2}(S^1 \times S^2, \tau)=1\).
(2) If \(N\) is the \(3\)-dimensional Klein bottle, then \(\hbox{ind}_{\mathbb Z_2}(S^1 \times S^2, \tau)=1\).
(3) If \(N\) is \(S^1 \times \mathbb R \mathbb P^2\), then \(\hbox{ind}_{\mathbb Z_2}(S^1 \times S^2, \tau)=2\), where \(\mathbb R \mathbb P^2\) denotes the projective plane.
\((4)\) If \(N\) is \(\mathbb R \mathbb P^3 \# \mathbb R \mathbb P^3\) which is the connected sum of projective spaces (sometimes, they are called lens spaces \(L(2,1)\)), then \(\hbox{ind}_{\mathbb Z_2}(S^1 \times S^2, \tau)=2\).
Note that the second theorem relies on the classification theorem by Y. Tao [Osaka Math. J. 14, 145–152 (1962; Zbl 0105.17302)]. More specifically, suppose \(f\) is a fixed-point free involution acting on \(S^1 \times S^2\). Then, the quotient (orbit) space \(N = S^1 \times S^2/ f\) is homeomorphic \(S^1 \times S^2\), \(S^1 \times \mathbb R \mathbb P^2\), \(\mathbb R \mathbb P^3 \# \mathbb R\mathbb P^3\) or \(3\)-dimensional Klein bottle.