Braided surfaces and their characteristic maps Louis Funar (joint - PowerPoint PPT Presentation

Characteristic maps of braided surfaces Elementary quotients of surface groups Lifting one step Profinite separability Braided surfaces and their characteristic maps Louis Funar (joint work with Pablo Pagotto) K-OS October 22, 2020 1 L.Funar Braided surfaces

Characteristic maps of braided surfaces Elementary quotients of surface groups Lifting one step Profinite separability Abstract We show that branched coverings of surfaces of large enough genus arise as characteristic maps of braided surfaces, thus being 2-prems. In the reverse direction we show that any nonabelian surface group has infinitely many finite simple nonabelian groups quotients with characteristic kernels which do not contain any simple loops and hence the quotient maps do not factor through free groups. By a pullback construction, finite dimensional Hermitian representations of braid groups provide invariants for the braided surfaces. We show that the strong equivalence classes of braided surfaces are separated by such invariants if and only if they are profinitely separated. 2 L.Funar Braided surfaces

Characteristic maps of braided surfaces Elementary quotients of surface groups Lifting one step Profinite separability Plan I. Characteristic maps for Braided surfaces II. Elementary quotients of surface groups III. Lifting one step IV. Profinite separability 3 L.Funar Braided surfaces

Characteristic maps of braided surfaces Elementary quotients of surface groups Lifting one step Profinite separability Let Σ denote a closed orientable surface. A braided surface over Σ is an embedding of a surface j : S → Σ × R 2 , such that the composition with the first factor projection j p → Σ × R 2 f : S ֒ → Σ is a branched covering. The composition p ◦ j is called the characteristic map of the braided surface S . Two braided surfaces j i : S → Σ × R 2 , i = 0 , 1 over Σ are equivalent if there exists some ambient isotopy h t : Σ × R 2 → Σ × R 2 , h 0 = id such that h t is fiber-preserving (i.e. there exists a homeomorphism ϕ : Σ → Σ such that p ◦ h t = ϕ ◦ p ) and h 1 ◦ j 0 = j 1 . When ϕ can be taken to be isotopic to the identity rel the branch locus, we say that the braided surfaces are strongly equivalent . Viro, Rudolph ’83, Kamada ’94, Carter-Kamada, Nakamura ’11 Edmonds ’99: f unramified covering, S contained in an orientable plane bundle, then its Euler class is torsion. 4 L.Funar Braided surfaces

Characteristic maps of braided surfaces Elementary quotients of surface groups Lifting one step Profinite separability Geometric Lifting Problem : When a ramified covering f : S → Σ lifts to a braided surface embedding ϕ : S → Σ × R 2 ? We could instead ask ϕ be an immersion and the embedding might be smooth, PL topologically flat, topological, etc. One might take f be a generic smooth/PL map and ask if it lifts to an embedding. Melikhov ’15. 5 L.Funar Braided surfaces

Characteristic maps of braided surfaces Elementary quotients of surface groups Lifting one step Profinite separability Two branched coverings f 0 , f 1 : S → Σ are equivalent if there exist homeomorphisms Φ : S → S and φ : Σ → Σ such that f 1 ◦ Φ = φ ◦ f 0 When φ is isotopic to the identity rel the branch locus, then the branched coverings are strongly equivalent . A degree n branched covering f : S → Σ of surfaces determines a holonomy homomorphism f ∗ : π 1 (Σ \ B ) → S n where B is the set of branch points. Hurwitz branched coverings Classification : Two branched coverings of surfaces are strongly equivalent if and only if their holonomy homomorphisms are conjugate . Moreover, they are equivalent if and only if the conjugacy classes of their holonomy homomorphisms are equivalent under the left action of the pure mapping class group Γ(Σ \ B ). 6 L.Funar Braided surfaces

Characteristic maps of braided surfaces Elementary quotients of surface groups Lifting one step Profinite separability A braided surfaces ϕ : S → Σ has degree n if its characteristic homomorphism f : S → Σ has degree n . A degree n braided surface ϕ : S → Σ × R 2 of surfaces determines a holonomy homomorphism ϕ ∗ : π 1 (Σ \ B ) → B n where B is the set of branch points of its characteristic map and B n is the braid group on n strands. Braided surfaces Classification : Two branched coverings of surfaces are strongly equivalent if and only if their holonomy homomorphisms are conjugate . Moreover, they are equivalent if and only if the conjugacy classes of their holonomy homomorphisms are equivalent under the left action of the pure mapping class group Γ(Σ \ B ). 7 L.Funar Braided surfaces

Characteristic maps of braided surfaces Elementary quotients of surface groups Lifting one step Profinite separability Our first result gives a positive answer to the geometric lifting problem, for large enough genus: Theorem There exists some h n,m such that every degree n branched covering S → Σ of a closed orientable surface Σ of genus g ≥ h n,m with at most n branch points occurs as the characteristic map of some braided surface. Petersen ’90 have proved that solvable unramified coverings can be lifted. 8 L.Funar Braided surfaces

Characteristic maps of braided surfaces Elementary quotients of surface groups Lifting one step Profinite separability Algebraic Lifting Problem Given a surjective group homomorphism p : � G → G , when does a homomorphism f : π 1 (Σ) → G lift to ϕ : π 1 (Σ) → � G ? We will restrict to surjective homomorphisms f and Σ will be a closed orientable surface. Definition The Schur class sc ( f ) ∈ H 2 ( G ) is the image f ∗ ([Σ]) of the fundamental class [Σ] of Σ. The action of Aut( π 1 (Σ) on Hom( π 1 (Σ) , G ) preserves the Schur classes. Moreover, the G -conjugacy acts trivially on H 2 ( G ). Thus the Schur class descends to a function: sc : Γ(Σ) \ Hom( π 1 (Σ) , G ) /G → H 2 ( G ) 9 L.Funar Braided surfaces

Characteristic maps of braided surfaces Elementary quotients of surface groups Lifting one step Profinite separability A homomorphism π 1 (Σ ′ ) → π 1 (Σ) is a pinch map if its is induced by the quotient (degree one) map Σ ′ → Σ which crushes several 1-handles to points. Definition A stabilization of f : π 1 (Σ) → G is the composition with a pinch map. Two homomorphisms are stably equivalent if they have stabilization equivalent under the Aut + ( π 1 (Σ) action. The stable equivalence descends also to G -conjugacy classes of homomorphisms. Observe that the image of a homomorphism is an invariant of its (stable) equivalence class. For this reason we shall restrict to surjective homomorphisms. Note that the Schur class of a homomorphism does not change under stabilization. 10 L.Funar Braided surfaces

Characteristic maps of braided surfaces Elementary quotients of surface groups Lifting one step Profinite separability Theorem (Livingston ’85, Zimmermann ’87) Surjective homomomorphisms are stably equivalent if and only if their Schur classes agree. If Ω n ( X ) is the dimension n orientable bordism group of X , then Thom proved that the natural map Ω n ( X ) → H n ( X ) is an isomorphism if n ≤ 3 and an epimorphism, if n ≤ 6. Two maps f : Σ → X, f ′ : Σ ′ → X representing the same class in H 2 ( X ) are therefore bordant and thus there exists a 3-manifold M 3 whose boundary is Σ ⊔ Σ ′ and a common extension F : M 3 → X . Consider a Heegaard surface Σ ′′ in M 3 , decomposing it into the union of two compression bodies C ∪ C ′ , glued together along their common boundary Σ ′′ by means of a homeomorphism ψ . A compression body is obtained from Σ ′′ × [0 , 1] by attaching 2-handles along disjoint nontrivial simple closed curves on Σ ′′ × { 1 } . Then F | Σ ′′ is a stabilization of f and f ′ , up to equivalence, throughout the restrictions F | C and F | C ′ . 11 L.Funar Braided surfaces

Characteristic maps of braided surfaces Elementary quotients of surface groups Lifting one step Profinite separability The stable algebraic lifting problem has a solution: Corollary Given a surjective p : � G → G , then a surjective homomorphism f : π 1 (Σ) → G lifts stably to � G if and only if there exists some class a ∈ H 2 ( � G ) such that p ∗ ( a ) = sc ( f ) . The Livingston-Zimmermann result was improved in the case when the target G is a finite group , as follows: Theorem (Dunfield-Thurston ’06) If G is finite, then there exists some g ( G ) such that every two surjective homomorphisms π 1 (Σ) → G with the same Schur class, for a closed orientable surface Σ of genus g ≥ g ( G ) , are equivalent. In particular, every such surjective homomorphism f : π 1 (Σ) → G lifts to � G , if there exists some a ∈ H 2 ( � G ) with p ∗ ( a ) = sc ( f ) . A key ingredient is that for large enough genus every surjective f should be a stabilization. 12 L.Funar Braided surfaces