Involutions on surfaces. (English) Zbl 1431.57026
This paper classifies all involutions on closed surfaces \(X\) up to isomorphism. The term involution means a map \(f: X \rightarrow X\) such that \(f^2 = id_X\). Suppose two involutions \(f\) and \(g\) are acting on \(X\) and there is a homeomorphism \(h: X \rightarrow X\). If \(h \circ f \circ h^{-1} = g\), then we say \(f\) and \(g\) are equivalent involutions or equivalent. In other words, \(f\) and \(g\) are in the same isomorphism class.
Significant reference sources are the works of T. Asoh [Hiroshima Math. J. 6, 171–181 (1976; Zbl 0332.57021)], J. Nielsen [Math.-Fys. Medd., Danske Vid. Selsk. 15, No. 1, 1–77 (1937; Zbl 0017.13302)], S. M. Natanzon [Russ. Math. Surv. 41, No. 5, 159–160 (1987; Zbl 0637.30040); translation from Usp. Mat. Nauk 41, No. 5(251), 191–192 (1986), correction ibid. 42, No. 2(254), 283 (1986); Russ. Math. Surv. 45, No. 6, 53–108 (1990; Zbl 0734.30037); translation from Usp. Mat. Nauk 45, No. 6(276), 47–90 (1990)], W. Scherrer [Comment. Math. Helv. 1, 69–119 (1929; JFM 55.0312.01)], P. A. Smith [Mich. Math. J. 14, 257–275 (1967; Zbl 0153.25402)], and E. Bujalance et al. [Math. Z. 211, No. 3, 461–478 (1992; Zbl 0758.30036)]. The case of orientable surfaces has been studied, which includes paper by F. Klein [Math. Ann. 42, 1–29 (1893; JFM 25.0689.03)]. However, the non-orientable case is much more complicated. In this sense, the author of this paper presents all results in an organized manner.
Let \(T_g\) denote the closed orientable surface of genus \(g\), and let \(N_r\) denote the connected sum of \(r\) copies of the projective planes \(\mathbb R \mathbb P^2\). Then, the following two statements are the main results of this article. Note that the number of the isomorphism classes includes the trivial map.
Theorem. The number of isomorphism classes of \(\mathbb Z_2\)-actions on \(T_g\) is equal to \(4 + 2g\).
Theorem. The number of isomorphism classes of \(\mathbb Z_2\)-actions on \(N_r\) is given by the following formulas:
\(\bullet\) if \(r \equiv 1 \pmod{4}\), then \(\frac{1}{64}(r^3 +9r^2+27r+91)\),
\(\bullet\) if \(r \equiv 3 \pmod{4}\), then \(\frac{1}{64}(r^3 +9r^2+23r+79)\),
\(\bullet\) if \(r \equiv 0 \pmod{4}\), then \(\frac{1}{64}(r^3 +18r^2+152)\),
\(\bullet\) if \(r \equiv 2 \pmod{4}\), then \(\frac{1}{64}(r^3 +18r^2+156r-8)\).
Consider \(T_0 = \mathbb S^2\), the two-sphere, since \(g = 0\). By the first theorem, there are four isomorphism classes of \(\mathbb Z_2\) on \(T_0 \). The four maps are: the identity, a \(180^\circ\)-rotation, the reflection at the “equator” line, and the antipodal map.
Consider \(N_1 = \mathbb R \mathbb P^2\). By the second theorem, there are two isomorphism classes of \(\mathbb Z_2\) on \(N_1\). These are: the identity and a \(180^\circ\)-rotation.
Note that the lifts of each map on \(N_1 = \mathbb R \mathbb P^2\) are shown the four isomorphism classes on \(T_0 = \mathbb S^2\). The identity and the antipodal maps on \(T_0 = \mathbb S^2\) both induce the trivial map on \(N_1 = \mathbb R \mathbb P^2\). However, the \(180^\circ\)-rotation and the reflection at the “equator” line induce a \(180^\circ\)-rotation on \(N_1 = \mathbb R \mathbb P^2\). J. Kalliongis and R. Ohashi discuss related topics in [Ars Math. Contemp. 15, No. 2, 297–321 (2018; Zbl 1414.57011)].
Significant reference sources are the works of T. Asoh [Hiroshima Math. J. 6, 171–181 (1976; Zbl 0332.57021)], J. Nielsen [Math.-Fys. Medd., Danske Vid. Selsk. 15, No. 1, 1–77 (1937; Zbl 0017.13302)], S. M. Natanzon [Russ. Math. Surv. 41, No. 5, 159–160 (1987; Zbl 0637.30040); translation from Usp. Mat. Nauk 41, No. 5(251), 191–192 (1986), correction ibid. 42, No. 2(254), 283 (1986); Russ. Math. Surv. 45, No. 6, 53–108 (1990; Zbl 0734.30037); translation from Usp. Mat. Nauk 45, No. 6(276), 47–90 (1990)], W. Scherrer [Comment. Math. Helv. 1, 69–119 (1929; JFM 55.0312.01)], P. A. Smith [Mich. Math. J. 14, 257–275 (1967; Zbl 0153.25402)], and E. Bujalance et al. [Math. Z. 211, No. 3, 461–478 (1992; Zbl 0758.30036)]. The case of orientable surfaces has been studied, which includes paper by F. Klein [Math. Ann. 42, 1–29 (1893; JFM 25.0689.03)]. However, the non-orientable case is much more complicated. In this sense, the author of this paper presents all results in an organized manner.
Let \(T_g\) denote the closed orientable surface of genus \(g\), and let \(N_r\) denote the connected sum of \(r\) copies of the projective planes \(\mathbb R \mathbb P^2\). Then, the following two statements are the main results of this article. Note that the number of the isomorphism classes includes the trivial map.
Theorem. The number of isomorphism classes of \(\mathbb Z_2\)-actions on \(T_g\) is equal to \(4 + 2g\).
Theorem. The number of isomorphism classes of \(\mathbb Z_2\)-actions on \(N_r\) is given by the following formulas:
\(\bullet\) if \(r \equiv 1 \pmod{4}\), then \(\frac{1}{64}(r^3 +9r^2+27r+91)\),
\(\bullet\) if \(r \equiv 3 \pmod{4}\), then \(\frac{1}{64}(r^3 +9r^2+23r+79)\),
\(\bullet\) if \(r \equiv 0 \pmod{4}\), then \(\frac{1}{64}(r^3 +18r^2+152)\),
\(\bullet\) if \(r \equiv 2 \pmod{4}\), then \(\frac{1}{64}(r^3 +18r^2+156r-8)\).
Consider \(T_0 = \mathbb S^2\), the two-sphere, since \(g = 0\). By the first theorem, there are four isomorphism classes of \(\mathbb Z_2\) on \(T_0 \). The four maps are: the identity, a \(180^\circ\)-rotation, the reflection at the “equator” line, and the antipodal map.
Consider \(N_1 = \mathbb R \mathbb P^2\). By the second theorem, there are two isomorphism classes of \(\mathbb Z_2\) on \(N_1\). These are: the identity and a \(180^\circ\)-rotation.
Note that the lifts of each map on \(N_1 = \mathbb R \mathbb P^2\) are shown the four isomorphism classes on \(T_0 = \mathbb S^2\). The identity and the antipodal maps on \(T_0 = \mathbb S^2\) both induce the trivial map on \(N_1 = \mathbb R \mathbb P^2\). However, the \(180^\circ\)-rotation and the reflection at the “equator” line induce a \(180^\circ\)-rotation on \(N_1 = \mathbb R \mathbb P^2\). J. Kalliongis and R. Ohashi discuss related topics in [Ars Math. Contemp. 15, No. 2, 297–321 (2018; Zbl 1414.57011)].
Reviewer: Ryo Ohashi (Wilkes-Barre)
MSC:
| 57M60 | Group actions on manifolds and cell complexes in low dimensions |
| 55M35 | Finite groups of transformations in algebraic topology (including Smith theory) |
Citations:
Zbl 0332.57021; Zbl 0017.13302; Zbl 0637.30040; Zbl 0734.30037; Zbl 0153.25402; Zbl 0758.30036; Zbl 1414.57011; JFM 55.0312.01; JFM 25.0689.03References:
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