--- title: Liftable Mapping Class Group url: https://www.emergentmind.com/topics/liftable-mapping-class-group type: topic --- # Liftable Mapping Class Group to=arxiv_search uppernarsjson {"query":"liftable mapping class group regular cyclic covers superelliptic branched covers torus disc algebraic mapping class group rigidity", "max_results": 10} A **liftable mapping class group** is the subgroup of a mapping class group consisting of isotopy classes of homeomorphisms on a base surface or orbifold that admit lifts through a specified covering map. In the formulations used across recent work, if \(p:S\to X\) is a finite-sheeted branched or unbranched cover, then a mapping class \(f\in \mathrm{Mod}(X)\) is liftable when some representative homeomorphism of \(f\) admits a lift to \(S\), and the corresponding subgroup is denoted \(\mathrm{LMod}_p(X)\) [2111.01626]. This notion appears in several closely related settings—regular cyclic covers of closed surfaces, cyclic branched covers of spheres, balanced superelliptic covers, branched torus covers, covers of surfaces with boundary, regular abelian covers, and branched covers of the disc—where liftability is detected by covering-theoretic, homological, or permutation-theoretic constraints. The subject is also closely linked to Birman–Hilden theory, which relates liftable mapping class groups downstairs to symmetric or normalizing mapping class groups upstairs [1804.10609]. ## 1. Definition and basic framework For a finite-sheeted regular branched cover \[ p:S_g\to S_{h,n}, \] the liftable mapping class group is defined by \[ \mathrm{LMod}_p(S_{h,n})=\{[f]\in \mathrm{Mod}(S_{h,n})\mid f \text{ admits a lift } \tilde f:S_g\to S_g \text{ with } p\circ \tilde f=f\circ p\} \] [2509.11788]. In the closed-surface setting, one likewise considers \(p_k:S_{k(g-1)+1}\to S_g\) and defines \(\mathrm{LMod}_{p_k}(S_g)\subset \mathrm{Mod}(S_g)\) as the set of mapping classes admitting lifts through \(p_k\) [1911.05682]. For covers of compact surfaces with boundary, the boundary-fixing condition is essential: a base homeomorphism may lift, but none of its lifts need fix the boundary pointwise, so liftability must be understood in the boundary-relative sense [1804.10609]. In the sphere-cover setting, the same concept is formulated for cyclic branched covers \(p:\Sigma_g\to \Sigma_0\) with branch set \(B\subset \Sigma_0\), where \[ LMod_p(\Sigma_0,B)=\{[f]\in Mod(\Sigma_0,B)\mid f \text{ lifts to a homeomorphism of } \Sigma_g\} \] [1604.03908]. Balanced superelliptic covers provide a particularly structured family: with \[ p_{g,k}:\Sigma_g\to \Sigma_0,\qquad g=n(k-1), \] the liftable groups are denoted \(\mathrm{LMod}_{2n+2;k}\), \(\mathrm{LMod}_{2n+2,\ast;k}\), and \(\mathrm{LMod}_{2n+1;k}^{1}\) in the closed, marked-point, and one-boundary variants, respectively [2203.13413, 2203.14460]. The Birman–Hilden perspective is central. For regular covers, one studies the symmetric mapping class group upstairs, consisting of fiber-preserving or deck-normalizing mapping classes. In several settings one has an exact sequence identifying the downstairs liftable group as a quotient of the upstairs symmetric group by the deck group, while in the boundary case the identification is direct: \[ SMod(\widetilde{\Sigma})\cong LMod(\Sigma,B) \] for finite-sheeted regular covers of compact surfaces with boundary, possibly branched [1804.10609]. For cyclic branched covers arising from a periodic \(F\in \map(S_g)\), one has \[ 1\to \langle F\rangle \to \smap_p(S_g)\to \lmap_p(S_{g_0,k})\to 1 \] and \( \smap_p(S_g)=N(F)\) [2308.12071]. ## 2. Birman–Hilden theory and symmetric groups upstairs A recurrent structural principle is that liftability downstairs corresponds to symmetry upstairs. For balanced superelliptic coverings, if \(\zeta\) is the balanced superelliptic rotation of order \(k\), the balanced superelliptic mapping class group consists of mapping classes represented by homeomorphisms satisfying \[ \varphi \langle \zeta\rangle \varphi^{-1}=\langle \zeta\rangle \] [2203.14460]. In that setting, for \(k\ge 3\), \[ \mathrm{LMod}_{2n+2} \cong SMod/\langle \zeta\rangle,\qquad \mathrm{LMod}_{2n+2,\ast} \cong SMod/\langle \zeta\rangle,\qquad \mathrm{LMod}_{2n+1}^{1}\cong SMod \] [2203.14460]. The analogous quotient statements also appear in the presentation-theoretic treatment of balanced superelliptic groups [2203.13413]. For periodic mapping classes \(F\) on closed surfaces, Broughton’s characterization yields exact sequences \[ 1 \to \langle F\rangle \to N(F)\to \map_{\Gamma_F}\to 1, \qquad 1 \to C(F)\to N(F)\to Z_n^\times(\Gamma_F)\to 1, \] and consequently \[ \lmap_p(S_{g_0,k})\cong N(F)/\langle F\rangle\cong \map_{\Gamma_F} \] [2308.12071]. This places liftable mapping class groups at the interface between branch-data combinatorics, periodic mapping classes, and normalizer/centralizer computations in \(\map(S_g)\). In the boundary case, the role of the fundamental group must be replaced by a fundamental groupoid. Choosing a finite set of basepoints \(A\subset \partial\Sigma^\circ\), the relevant object is \[ G=\pi_1(\Sigma^\circ,A), \] together with a subgroupoid \(H\) coming from the cover. The subgroup \(LAut_H(G)\subset PAut(G)\) is defined by preserving \(H\) and acting trivially on the quotient groupoid \(G/H\), and the groupoid Birman–Hilden theorem gives \[ \Pi:SAut(K)\to LAut_H(G)\text{ is an isomorphism} \] [1804.10609]. This algebraic replacement is necessary precisely because boundary-fixing liftability is not detected by \(\pi_1\) alone. ## 3. Criteria for liftability A major theme of the subject is that liftability admits explicit algebraic criteria in concrete covering families. For the standard regular \(k\)-sheeted cyclic cover \[ p_k:S_{k(g-1)+1}\to S_g, \] liftability is detected on mod-\(k\) homology by the symplectic representation \[ \Psi:\mathrm{Mod}(S_g)\to \mathrm{Sp}(2g,\mathbb Z),\qquad \Psi_k. \] The following are equivalent for \(f\in \mathrm{Mod}(S_g)\): \[ f\in \mathrm{LMod}_{p_k}(S_g), \] \[ f\in \mathrm{Stab}_{\mathrm{Mod}(S_g)}(\{\ell e_1:\ell\in \mathbb Z_k^\times\}), \] and \[ \Psi_k(f)=(e_{ij})\quad\text{with}\quad e_{2i}=0\ (i\neq 2),\qquad e_{22}\in \mathbb Z_k^\times \] [1911.05682, 2111.01626]. Equivalently, if \(\Psi(f)=(d_{ij})\), then \[ k\mid d_{2i}\quad\text{for all }i\neq 2,\qquad \gcd(d_{22},k)=1 \] [1911.05682]. This identifies \(\mathrm{LMod}_{p_k}(S_g)\) as the subgroup preserving the cyclic set of primitive multiples of \(e_1\) in \(H_1(S_g,\mathbb Z_k)\) [2111.01626]. For the torus-branched covers \[ p_k:S_k\to S_{1,2}, \] the relevant representation is \[ \Psi:\mathrm{Mod}(S_{1,2})\to \mathrm{GL}_3(\mathbb Z), \] and the homological image of the liftable subgroup is \[ \Psi(\mathrm{LMod}_{p_k}(S_{1,2}))= \left\{ \begin{pmatrix} A&0\\ v&\pm 1 \end{pmatrix} \middle| A\in \mathrm{SL}_2(\mathbb Z),\ v\in k\mathbb Z\times k\mathbb Z \right\} \] [2509.11788]. Thus liftability is governed by a congruence condition on the lower-left vector, and \[ [\mathrm{Mod}(S_{1,2}):\mathrm{LMod}_{p_k}(S_{1,2})]=k^2 \] [2509.11788]. For balanced superelliptic covers, liftability is characterized by parity of the induced permutation on branch points. If \[ \Psi:\mathrm{Mod}(\Sigma_0;B,\emptyset)\to S_{2n+2} \] is the puncture-permutation map and \(W_{2n+2}\subset S_{2n+2}\) is the subgroup of permutations that are either parity-preserving or parity-reversing, then \[ \mathrm{LMod}_{2n+2}=\Psi^{-1}(W_{2n+2}) \] [2203.14460, 2203.13413]. In the corresponding older formulation for balanced superelliptic covers, the image is the subgroup \(W_{2n+2}\subset S_{2n+2}\) consisting of permutations that either preserve parity or reverse parity, giving \[ 1\to PMod(\Sigma_0,B)\to LMod_{g,k}(\Sigma_0,B)\to W_{2n+2}\to 1 \qquad (k\ge 3) \] [1604.03908]. A finer curve criterion is also available in this family: \[ \gamma \text{ lifts } \iff \hat i(\gamma,\alpha)\equiv 0 \pmod{k}, \] and in homology \[ \gamma \text{ lifts } \iff \sum_{i=1}^{2n+1}(-1)^{i+1}\gamma_i\equiv 0\pmod{k} \] [1604.03908]. For regular abelian covers \(p:S_g\to S_h\) with deck group \(A\), liftability reduces to preserving the kernel of the induced homology map \[ \eta' : H_1(S_h;\mathbb Z)\to A, \] namely \[ F\in \lmap_p(S_h)\quad \Longleftrightarrow\quad F_{\#}(\ker \eta')=\ker \eta' \] [2412.07319]. This homological characterization underlies the algorithmic treatment of regular abelian covers. ## 4. Structural properties: index, normal series, maximality, and rigidity phenomena Liftable mapping class groups are typically finite-index subgroups of ambient mapping class groups. For finite-sheeted branched covers of hyperbolic surfaces, \(\mathrm{LMod}_p(X)\) is finite-index in \(\mathrm{Mod}(X)\) [2111.01626]. In the boundary setting, \(LMod(\Sigma,B)\) is always finite-index in \(Mod(\Sigma,B)\) [1804.10609]. For torus-branched covers \(p_k:S_k\to S_{1,2}\), the exact index is \(k^2\) [2509.11788]. For balanced superelliptic covers with \(k>2\), \[ [Mod(\Sigma_0,B):LMod_{g,k}(\Sigma_0,B)] =\frac{(2n+2)!}{2((n+1)!)^2} \] [1604.03908]. The cyclic closed-surface covers admit a congruence-type normal series \[ 1 \lhd \mathrm{Mod}(S_g)[k] \lhd \mathrm{Mod}_{p_k}(S_g,e_1) \lhd \mathrm{LMod}_{p_k}(S_g), \] with \[ \mathrm{LMod}_{p_k}(S_g)/\mathrm{Mod}_{p_k}(S_g,e_1)\cong \mathbb Z_k^\times \] [1911.05682, 2111.01626]. This is presented as a higher-genus analog of \[ 1\lhd \Gamma(k)\lhd \Gamma_1(k)\lhd \Gamma_0(k)\subset \mathrm{SL}(2,\mathbb Z) \] [1911.05682]. For the infinite ladder cover \(q_g:\mathcal L\to S_g\), \[ UMod(S_g):=\bigcap_{k\ge 2}\mathrm{LMod}_{p_k}(S_g)=\mathrm{LMod}_{q_g}(S_g) \] [2111.01626]. Several families exhibit strong subgroup-theoretic behavior. For \(g\ge 2,\ k\ge 2\), \[ N_{\mathrm{Mod}(S_g)}(\mathrm{LMod}_{p_k}(S_g))=\mathrm{LMod}_{p_k}(S_g), \] so the cyclic-cover liftable groups form an infinite family of self-normalizing subgroups, and if \(k\) is prime they are maximal in \(\mathrm{Mod}(S_g)\) [2111.01626]. The torus-branched family satisfies the parallel criterion \[ \mathrm{LMod}_{p_k}(S_{1,2}) \text{ is maximal in } \mathrm{Mod}(S_{1,2}) \iff k \text{ is prime} \] [2509.11788]. At the opposite extreme, some covers have every mapping class lift. For finite-sheeted regular covers of compact surfaces with boundary, possibly branched, one has \[ LMod(\Sigma,B)=Mod(\Sigma,B) \] if and only if the cover is a Burau cover [1804.10609]. In the hyperelliptic case \(k=2\), for cyclic branched sphere covers, \[ LMod_p(\Sigma_0,B)=Mod(\Sigma_0,B) \] [1604.03908]. More generally, for spherical cyclic data, one recovers the criterion \[ \lmap_p(S_{0,k})=\map(S_{0,k}) \quad\Longleftrightarrow\quad c_1=\cdots=c_k\ \text{ and }\ k\equiv 0\pmod n \] [2308.12071]. ## 5. Generating sets and presentations A prominent development in the subject is the derivation of explicit finite generating sets. For regular cyclic covers of closed surfaces with \(g\ge 3\), \(k\ge 2\), \[ \mathrm{LMod}_{p_k}(S_g) = \left\langle S_k'' \cup \tilde{S}_k \cup S_{g,k} \cup \{T_{a_2},T_{b_2}, \ldots,T_{a_g},T_{b_g},T_{c_1},\ldots,T_{c_{g-1}\} \right\rangle \] [2111.01626]. Here \(S_k''\) consists of explicit coset representatives, \(\tilde S_k\) is a lift of a finite generating set for \(\Gamma_1(k)\subset SL(2,\mathbb Z)\), and \(S_{g,k}=\{F_1,\dots,F_{g-2}\}\) is a collection of bounding pair maps [2111.01626]. The genus-\(2\), degree-\(2\) exceptional case is \[ \mathrm{LMod}_{p_2}(S_2) = \langle T_{a_1}, T_{b_1}^2, T_{c_1}, T_{a_2}, T_{b_2} \rangle \] [2111.01626]. For cyclic branched covers of \(S_{1,2}\), the uniform theorem is \[ \mathrm{LMod}_{p_k}(S_{1,2}) = \left\langle T_a,\ T_b,\ T_c^k,\ \iota,\ (T_bT_c)^6,\ T_c^jT_b^iT_a(T_bT_c)^6T_a^{-1}T_b^{-i}T_c^{-j} \ \middle|\ 1\le i,j