Simplified presentations for non-orientable surface mapping class groups.
problem Presenting the mapping class group of non-orientable surfaces.
method Provided simpler infinite presentations.
result Simplified presentations for the group.
Characteristic classes of oriented vector bundles can be identified with cohomology classes of the disjoint union of classifying spaces BSO_n of special orthogonal groups SO_n with n=0,1,... A characteristic class is stable if it extends to a cohomology class of a homotopy colimit BSO of classifying spaces BSO_n. Simil…
This paper provides an infinite presentation for a subgroup of mapping class groups of non-orientable surfaces.
problem Finite presentations for mapping class groups of non-orientable surfaces.
method Using Stukow's finite presentation and Birman exact sequences.
result An infinite presentation for the twist subgroup of the mapping class group of a compact non-orientable surface.
Study on dimensions of mapping class groups of non-orientable surfaces.
problem Determining dimensions of mapping class groups for non-orientable surfaces.
method Analyzing cohomological, geometric, and virtual dimensions.
result Equal dimensions for Ng when geq4,5. Study on Euler class and flux homomorphisms for non-orientable surfaces.
problem Investigate Euler class and flux homomorphisms for non-orientable surfaces.
method Analyze Euler class and flux homomorphisms for non-orientable compact surfaces with one boundary component.
result Prove the simplicity of the kernel of the flux homomorphisms, implying the non-existence of invariants analogous to the Calabi invariant.
Presented an infinite group presentation for non-orientable surfaces.
problem Presenting the mapping class group of non-orientable surfaces.
method Generalized an existing presentation for mapping class groups.
result An infinite presentation for non-orientable surfaces with boundary.
Study homology groups for non-orientable surfaces with boundary.
problem Determine the first homology group for mapping class groups of non-orientable surfaces.
method Calculate the first homology group with twisted coefficients.
result Explicitly determined the first homology group for various mapping class groups.
The paper generalizes conditions for Euler characteristic and orientability of manifolds.
problem Conditions for Euler characteristic and orientability of manifolds.
method Generalization of known conditions for Euler characteristic and orientability of manifolds.
result Even Euler characteristic for k-orientable manifolds unless dimension is a multiple of 2k+1. Study Farrell cohomology for non-orientable surfaces, classifying subgroup conjugacy.
problem Determine the p-primary component of Farrell cohomology for non-orientable surfaces. method Classify subgroups of order p using topological equivalence adapted to surfaces with marked points. result Determine the p-primary component of Farrell cohomology for non-orientable surfaces. Birman-Lubotzky-McCarthy proved that any abelian subgroup of the mapping class groups for orientable surfaces is finitely generated. We apply Birman-Lubotzky-McCarthy's arguments to the mapping class groups for non-orientable surfaces. We especially find a finitely generated group isomorphic to a given torsion-free sub…
In this paper we study the minimum dilatation pseudo-Anosov mapping classes coming from fibrations over the circle of a single 3-manifold, the mapping torus for the "simplest pseudo-Anosov braid". The dilatations that arise include the minimum dilatations for orientable mapping classes for genus g=2,3,4,5,8 as well as …
Study calculates integral cohomology of non-orientable infinite type surfaces.
problem Computing the first integral cohomology group of non-orientable infinite type surfaces.
method Alexander method, isomorphism to automorphism group, topological rigidity of curve graph, semi-direct product structure.
result First integral cohomology group computed for non-orientable infinite type surfaces.
Finite presentations for the mapping class group M(F) are known for arbitrary orientable compact surface F. If F is non-orientable, then such presentations are known only when F has genus at most 3 and few boundary components. In this paper we obtain finite presentation for the mapping class group of the closed non-ori…
Study on achiral Sol 3-manifolds with density results.
problem Understanding the achirality of commensurable classes of Sol 3-manifolds.
method Investigated commensurable classes of Sol 3-manifolds using discriminant invariants.
result Infinitely many achiral commensurable classes, but their density is 0.
Study the first homology group for a specific mapping class group.
problem Calculating the first homology group for a specific mapping class group.
method Determined using coefficients in H1(N;Z) for a non-orientable surface. result Determined the first homology group for the mapping class group of a specific surface.
Method calculates Z2-Thurston norm for Seifert 3-manifolds using pseudo-horizontal surfaces.
problem Computing Z2-Thurston norm for Z2-homology classes in orientable Seifert 3-manifolds. method Using pseudo-horizontal surfaces, we provide a necessary and sufficient criterion for their existence, calculate non-orientable genera, and develop an algorithm to compute the Z2-Thurston norm. result We successfully compute the Z2-Thurston norm for every Z2-homology class in orientable Seifert 3-manifolds. Characterizes closures of mapping class group orbits on non-orientable surfaces.
problem Understanding closures of orbits in Teichmüller spaces for non-orientable surfaces.
method Analyzes closures in ML and PML for measured laminations, projective measured laminations, and points. result Characterizes closures of weighted two-sided curves in ML. A spine is constructed for a non-orientable surface's decorated Teichmüller space.
problem Constructing a spine for a non-orientable surface's decorated Teichmüller space.
method Building on Harer's work, constructing a spine and computing its dimension, showing equivariance with the pure mapping class group.
result A spine is constructed with minimal dimension for a punctured non-orientable surface.
A {\em word labeled oriented graph} (WLOG) is an oriented graph G on vertices X={x1,…,xk}, where each oriented edge is labeled by a word in X±1. WLOGs give rise to presentations which generalize Wirtinger presentations of knots. WLOG presentations, where the underlying graph is a tree are of …
Knots in 3-manifolds are equivalent if isotopic, except in special cases.
problem Understanding when knots in 3-manifolds are equivalent and isotopic.
method Analyzing prime, closed, oriented 3-manifolds and irreducible manifolds, and considering orientation-preserving mapping class groups and homeomorphisms.
result Knots in prime, closed, oriented 3-manifolds are isotopic if and only if the orientation preserving mapping class group is trivial.
Study shows torsion normal generators in non-orientable surface mapping class groups.
problem Understanding normal generators of mapping class groups for non-orientable surfaces.
method Analyzes periodic elements and their normal closures in non-orientable surfaces.
result Provides conditions for normal generation and examples of elements.
We construct several families of embeddings of braid groups into mapping class groups of orientable and non-orientable surfaces and prove that they induce the trivial map in stable homology in the orientable case, but not so in the non-orientable case. We show that these embeddings are non-geometric in the sense that t…
Classifies manifolds with dense conjugacy classes in their mapping class groups.
problem Classifying manifolds based on conjugacy classes in their mapping class groups.
method Analyzing connected orientable 2-manifolds and their mapping class groups.
result Mapping class groups of certain manifolds have dense conjugacy classes.
J. Boyle classified 1-handles attached to surface-knots, that are closed and connected surfaces embedded in the Euclidean 4-space, in the case that the surfaces are oriented and 1-handles are orientable with respect to the orientations of the surfaces. In that case, the equivalence classes of 1-handles correspond to th…
The paper constructs 3-manifolds with co-orientable taut foliations but no foliations with vanishing Euler class.
problem Constructing 3-manifolds with co-orientable taut foliations but no foliations with vanishing Euler class.
method Constructs infinitely many rational homology 3-spheres using Dehn surgeries and Heegaard Floer homology.
result Found rational homology 3-spheres that admit co-orientable taut foliations but none with vanishing Euler class.
Study non-orientable surfaces in gauge theories, matching fluxes with homotopy classes.
problem Understanding non-orientable surfaces in gauge theory and their electric-magnetic duality.
method Reduction of twisted N=4 gauge theory on non-orientable manifolds, matching fluxes with homotopy classes.
result Verification of mirror symmetry of branes as predicted by S-duality.
Study shows mapping class group dimension for surfaces with punctures.
problem Determining the geometric dimension of mapping class groups of surfaces with punctures.
method Proved cocompact classifying space for proper actions with dimension equal to virtual cohomological dimension.
result Proper geometric dimension of mapping class groups of orientable surfaces with punctures.
Paper finds a minimal set of Dehn twists to generate twist subgroup of non-orientable surfaces.
problem Finding a minimal set of Dehn twists to generate the twist subgroup of non-orientable surfaces.
method Used Dehn twists and an argument from Hirose [5] to find a minimal generating set.
result Found a generating set with one less element than the lower bound.
The thesis proves a subgroup of non-orientable surfaces is generated by six involutions.
problem Identifying the generating set of a subgroup in non-orientable surfaces.
method Analyzing Dehn twists and involutions to prove the subgroup generation.
result The twist subgroup Tg is generated by six involutions for g≥16 or g=14. The paper studies finite group actions on Seifert manifolds, proving conditions for their existence.
problem Finite group actions on Seifert manifolds that preserve fiber and orientation.
method Constructs group actions and refines obstruction conditions.
result Provides general structure of finite groups that can act on Seifert manifolds.
The study explores the structure of mapping class groups of non-orientable surfaces.
problem Understanding the structure of mapping class groups of non-orientable surfaces.
method Explains relations and provides generating sets for the level d mapping class groups.
result Normal and finite generating sets for the mapping class groups are provided.
Nonorientable surface mapping class group embeds quasi-isometrically in its orientable cover.
problem Embedding nonorientable surface mapping class group in orientable surface mapping class group.
method Utilized semihyperbolicity of orientable surface mapping class group and orientation double covering properties.
result Injective homomorphism is a quasi-isometric embedding.
Study restricts causal graphs with expert knowledge.
problem Restricting causal graphs to include expert orientation knowledge.
method Prove properties, present new orientation rules, develop algorithms.
result Shows how to uniquely represent restricted essential ancestral graphs.
Classifies isotopy classes of links from Thompson's group F and its subgroup.
problem Classify isotopy classes of links from Thompson's group F and its subgroup.
method Introduced a method to produce links from elements of Thompson's group F and its subgroup.
result Classified isotopy classes of links from Thompson's group F and its subgroup.
Study on mapping class groups of non-orientable surfaces, proving some conjectures and refuting others.
problem Analogies between Fuchsian groups and mapping class groups of non-orientable surfaces.
method Analyzing limit sets, foliations, and geometric properties.
result Established parts of a conjecture about the limit set and provided evidence for and against the analogy.
Let M be a compact closed non-orientable surface. We show that the space of representations of the fundamental group of M into PSL(2,R) has exactly two connected components. These two components are the preimages of a certain Stiefel-Whitney characteristic class, computed in a similar way as the Euler class in the orie…
We prove that the homology of the mapping class groups of non-orientable surfaces stabilizes with the genus of the surface. Combining our result with recent work of Madsen and Weiss, we obtain that the classifying space of the stable mapping class group of non-orientable surfaces, up to homology isomorphism, is the inf…
Study algebraic K-theory for specific groups of non-orientable surfaces.
problem Algebraic K-theory of group rings for specific non-orientable surface groups.
method Detailed analysis of group rings and algebraic K-theory.
result General formula for algebraic K-theory groups of mapping class groups of non-orientable surfaces.
Injective homomorphisms of non-orientable mapping class groups are conjugate.
problem Characterizing conjugacy classes of injective homomorphisms in non-orientable mapping class groups.
method Proving conjugacy of injective homomorphisms by showing existence of a fixed element.
result Abstract commensurator of non-orientable mapping class groups equals the group itself.
Paper introduces class-oriented poisoning attacks to improve adversarial availability attacks.
problem Improving adversarial availability attacks on neural networks.
method Gradient-based framework for crafting poisoned images at the class level.
result Demonstrated effectiveness of class-oriented poisoning attacks on various models and datasets.
By considering appropriate finite covering spaces of closed non-orientable surfaces, we construct linear representations of their mapping class group which have finite index image in certain big arithmetic groups.
Study fiber-preserving, orientation-reversing involutions on Seifert fibered 3-manifolds.
problem Characterize fiber-preserving, orientation-reversing involutions on Seifert fibered 3-manifolds.
method Construct a class of involutions, extend product involutions, and use double covering.
result Any fiber-preserving, orientation-reversing involution factors as a product of an orientation-preserving and a specific class of involutions.
Different directed acyclic graphs (DAGs) may be Markov equivalent in the sense that they entail the same conditional independence relations among the observed variables. Meek (1995) characterizes Markov equivalence classes for DAGs (with no latent variables) by presenting a set of orientation rules that can correctly i…
For a closed oriented 3-manifold Y, we define an absolute grading on the Heegaard Floer homology groups of Y by homotopy classes of oriented 2-plane fields. We show that this absolute grading refines the relative one and that it is compatible with the maps induced by cobordisms. We also prove that if ξ is a contact str…
The present paper are the notes of a mini-course addressed mainly to non-experts. It purpose it to provide a first approach to the theory of mapping class groups of non-orientable surfaces.
We give an infinite presentation for the mapping class group of a non-orientable surface. The generating set consists of all Dehn twists and all crosscap pushing maps along simple loops.
We give a finite presentation of the mapping class group of an oriented (possibly bounded) surface of genus greater or equal than 1, considering Dehn twists on a very simple set of curves.
Study crosscap transpositions on non-orientable surfaces, providing explicit formulae.
problem Understanding the action of crosscap transpositions on multicurves in non-orientable surfaces.
method Explicit formulae for crosscap transpositions and their inverses on multicurves in Ng,n. result Explicit formulae for crosscap transpositions and their inverses on multicurves in non-orientable surfaces.