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European Mathematical Society Publishing House
2024-03-28 19:45:54
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https://www.ems-ph.org/meta/jmeta-stream.php?jrn=CMH&vol=85&iss=1&update_since=2024-03-28
Commentarii Mathematici Helvetici
Comment. Math. Helv.
CMH
0010-2571
1420-8946
General
10.4171/CMH
http://www.ems-ph.org/doi/10.4171/CMH
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European Mathematical Society Publishing House
Zuerich, Switzerland
© Swiss Mathematical Society
85
2010
1
Topological rigidity and Gromov simplicial volume
Pierre
Derbez
Université de Provence, MARSEILLE CEDEX 13, FRANCE
Haken manifold, Seifert fibered space, hyperbolic 3-manifold, Gromov simplicial volume, non-zero degree maps, Dehn filling, subgroup separability
A natural problem in the theory of 3-manifolds is the question of whether two 3-manifolds are homeomorphic or not. The aim of this paper is to study this problem for the class of closed Haken manifolds using degree one maps. To this purpose we introduce an invariant τ(N) = (Vol(N),||N||), where ||N|| denotes the Gromov simplicial volume of N and Vol(N) is a 2-dimensional simplicial volume which measures the volume of the base 2-orbifolds of the Seifert pieces of N. After studying the behavior of τ(N) under the action of non-zero degree maps, we prove that if M and N are closed Haken manifolds such that ||M|| = |deg(f)| ||N|| and Vol(M) = Vol(N) then any non-zero degree map f: M → N is homotopic to a covering map. As a corollary we prove that if M and N are closed Haken manifolds such that τ(N) is sufficiently close to τ(M) then any degree one map f: M → N is homotopic to a homeomorphism.
Manifolds and cell complexes
Geometry
General
1
37
10.4171/CMH/186
http://www.ems-ph.org/doi/10.4171/CMH/186
On minimal surfaces bounded by two convex curves in parallel planes
Martin
Traizet
Université François Rabelais, TOURS, FRANCE
Minimal surface with boundary, Robin function, concentration of curvature
We prove that a compact minimal surface bounded by two closed convex curves in parallel planes close enough to each other must be topologically an annulus.
Differential geometry
General
39
71
10.4171/CMH/187
http://www.ems-ph.org/doi/10.4171/CMH/187
Foliations and global inversion
Eduardo
Cabral Balreira
Trinity University, SAN ANTONIO, UNITED STATES
Jacobian conjecture, Hadamard’s theorem, global inversion theorem
We consider topological conditions under which a locally invertible map admits a global inverse. Our main theorem states that a local diffeomorphism f: M → ℝn is bijective if and only if Hn − 1(M) = 0 and the pre-image of every affine hyperplane is non-empty and acyclic. The proof is based on some geometric constructions involving foliations and tools from intersection theory. This topological result generalizes in finite dimensions the classical analytic theorem of Hadamard–Plastock, including its recent improvement by Nollet–Xavier. The main theorem also relates to a conjecture of the aforementioned authors, involving the well-known Jacobian conjecture in algebraic geometry.
Global analysis, analysis on manifolds
Manifolds and cell complexes
General
73
93
10.4171/CMH/188
http://www.ems-ph.org/doi/10.4171/CMH/188
Monodromy in Hamiltonian Floer theory
Dusa
McDuff
Columbia University, NEW YORK, UNITED STATES
Hamiltonian group, Floer theory, Seidel representation, spectral invariant, Hofer norm, Calabi quasimorphism
Schwarz showed that when a closed symplectic manifold (M,ω) is symplectically aspherical (i.e. the symplectic form and the first Chern class vanish on π2(M)) then the spectral invariants, which are initially defined on the universal cover of the Hamiltonian group, descend to the Hamiltonian group Ham(M,ω). In this note we describe less stringent conditions on the Chern class and quantum homology of M under which the (asymptotic) spectral invariants descend to Ham(M,ω). For example, they descend if the quantum multiplication of M is undeformed and H2(M) has rank > 1, or if the minimal Chern number is at least n + 1 (where dim M = 2n) and the even cohomology of M is generated by divisors. The proofs are based on certain calculations of genus zero Gromov–Witten invariants. As an application, we show that the Hamiltonian group of the one point blow up of T4 admits a Calabi quasimorphism. Moreover, whenever the (asymptotic) spectral invariants descend it is easy to see that Ham(M,ω) has infinite diameter in the Hofer norm. Hence our results establish the infinite diameter of Ham in many new cases. We also show that the area pseudonorm – a geometric version of the Hofer norm – is nontrivial on the (compactly supported) Hamiltonian group for all noncompact manifolds as well as for a large class of closed manifolds.
Differential geometry
Manifolds and cell complexes
General
95
133
10.4171/CMH/189
http://www.ems-ph.org/doi/10.4171/CMH/189
Virtually soluble groups of type FP∞
Conchita
Martínez-Pérez
Universidad de Zaragoza, ZARAGOZA, SPAIN
Brita E. A.
Nucinkis
Royal Holloway University of London, EGHAM, SURREY, UNITED KINGDOM
Cohomological finiteness conditions, soluble groups
We prove that a virtually soluble group G of type FP∞ admits a finitely dominated model for EG of dimension the Hirsch length of G. This implies in particular that the Brown conjecture is satisfied for virtually torsion-free elementary amenable groups.
Group theory and generalizations
General
135
150
10.4171/CMH/190
http://www.ems-ph.org/doi/10.4171/CMH/190
SLn(ℤ[t]) is not FPn − 1
Kai-Uwe
Bux
Universität Bielefeld, BIELEFELD, GERMANY
Amir
Mohammadi
The University of Texas at Austin, AUSTIN, UNITED STATES
Kevin
Wortman
University of Utah, SALT LAKE CITY, UNITED STATES
Euclidean buildings, finiteness properties, geometric group theory
We prove the result from the title using the geometry of Euclidean buildings.
Group theory and generalizations
General
151
164
10.4171/CMH/191
http://www.ems-ph.org/doi/10.4171/CMH/191
Reconstructing p-divisible groups from their truncations of small level
Adrian
Vasiu
Binghamton University, BINGHAMTON, UNITED STATES
p-divisible groups, F-crystals, algebras, affine group schemes
Let k be an algebraically closed field of characteristic p > 0. Let D be a p-divisible group over k. Let nD be the smallest non-negative integer for which the following statement holds: if C is a p-divisible group over k of the same codimension and dimension as D and such that C[pnD] is isomorphic to D[pnD], then C is isomorphic to D. To the Dieudonné module of D we associate a non-negative integer ℓD which is a computable upper bound of nD. If D is a product ∏i ∈ I Di of isoclinic p-divisible groups, we show that nD = ℓD; if the set I has at least two elements we also show that nD ≤ max{1, nDi, nDi + nDj − 1 | i, j ∈ I, j ≠ i}. We show that we have nD ≤ 1 if and only if ℓD ≤ 1; this recovers the classification of minimal p-divisible groups obtained by Oort. If D is quasi-special, we prove the Traverso truncation conjecture for D. If D is F-cyclic, we explicitly compute nD. Many results are proved in the general context of latticed F-isocrystals with a (certain) group over k.
Number theory
Algebraic geometry
Group theory and generalizations
General
165
202
10.4171/CMH/192
http://www.ems-ph.org/doi/10.4171/CMH/192
Contact homology of Hamiltonian mapping tori
Oliver
Fabert
Universität München, MÜNCHEN, GERMANY
Symplectic field theory, Floer homology, J-holomorphic curve
In the general geometric setup for symplectic field theory the contact manifolds can be replaced by mapping tori MΦ of symplectic manifolds (M,ω) with symplectomorphisms Φ. While the cylindrical contact homology of MΦ is given by the Floer homologies of powers of Φ, the other algebraic invariants of symplectic field theory for MΦ provide natural generalizations of symplectic Floer homology. For symplectically aspherical M and Hamiltonian Φ we study the moduli spaces of rational curves and prove a transversality result, which does not need the polyfold theory by Hofer, Wysocki and Zehnder. We use our result to compute the full contact homology of MΦ ≅ S1 × M.
Differential geometry
General
203
241
10.4171/CMH/193
http://www.ems-ph.org/doi/10.4171/CMH/193