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Group of isotopy classes of a topological automorphism group
the mapping class group is an important algebraic invariant of a topological space. Briefly, the mapping class group is a certain discrete group corresponding
Mapping_class_group
Concept in mathematics
precisely in topology, the mapping class group of a surface, sometimes called the modular group or Teichmüller modular group, is the group of homeomorphisms of
Mapping class group of a surface
Mapping_class_group_of_a_surface
Term in geometric topology
theorem of Max Dehn that maps of this form generate the mapping class group of isotopy classes of orientation-preserving homeomorphisms of any closed,
Dehn_twist
Group whose operation is a composition of braids
resulting group each of whose completion yields a different group. The first is a very tame group and is isomorphic to the mapping class group of the infinitely
Braid_group
Characterizes homeomorphisms of a compact orientable surface
of the mapping class group Mod(S). In fact, the proof of the classification theorem leads to a canonical representative of each mapping class with good
Nielsen–Thurston classification
Nielsen–Thurston_classification
topological group. In geometric topology especially, one considers the quotient group obtained by quotienting out by isotopy, called the mapping class group: M
Homeomorphism_group
Parametrizes complex structures on a surface
he used in his study of the mapping class group of a surface. Other more combinatorial objects associated to this group (in particular the curve complex)
Teichmüller_space
Isomorphism of differentiable manifolds
Its component group is called the mapping class group. In dimension 2 (i.e. surfaces), the mapping class group is a finitely presented group generated by
Diffeomorphism
Mathematical monograph on braid groups
Braids, Links, and Mapping Class Groups is a mathematical monograph on braid groups and their applications in low-dimensional topology. It was written
Braids, Links, and Mapping Class Groups
Braids,_Links,_and_Mapping_Class_Groups
Type of mathematical group
explicit matrices. The mapping class group of a genus 2 surface is also known to be linear. In some cases the fundamental group of a manifold can be shown
Linear_group
Doughnut-shaped surface of revolution
The homeomorphism group (or the subgroup of diffeomorphisms) of the torus is studied in geometric topology. Its mapping class group (the connected components
Torus
Mathematical group
group is important in the topology of surfaces because there is a connection provided by the Dehn–Nielsen theorem: the extended mapping class group of
Outer_automorphism_group
Orientation-preserving mapping class group of the torus
Henrik Abel in 1827. Bianchi group Classical modular curve Fuchsian group J-invariant Kleinian group Mapping class group Minkowski's question-mark function
Modular_group
Three-holed sphere
the Farey graph. The action of the mapping class group on the pants complex is of interest for studying this group. For example, Allen Hatcher and William
Pair_of_pants_(mathematics)
sphere of dimension 6 g − 7 {\displaystyle 6g-7} . The action of the mapping class group on the Teichmüller space extends continuously over the union with
Thurston_boundary
the study of the geometry of the Teichmüller space, of mapping class groups and of Kleinian groups. It was introduced by W.J.Harvey in 1978. Let S {\displaystyle
Curve_complex
automorphisms is the outer automorphism group of a free group, which is similar in some ways to the mapping class group of a surface. Jakob Nielsen (1924)
Automorphism group of a free group
Automorphism_group_of_a_free_group
Function, homomorphism, or morphism
map or mapping is a function in its general sense.[vague] These terms may have originated as from the process of making a geographical map: mapping the Earth
Map_(mathematics)
Mapping which preserves all topological properties of a given space
between two continuous functions Mapping class group – Group of isotopy classes of a topological automorphism group Poincaré conjecture – Theorem in geometric
Homeomorphism
One-dimensional complex manifold
the marking) one takes the quotient of Teichmüller space by the mapping class group. In this case it is the modular curve. In the remaining cases, X
Riemann_surface
Decomposition of a compact oriented 3-manifold by dividing it into two handlebodies
specified up to taking a double coset in the mapping class group of H. This connection with the mapping class group was first made by W. B. R. Lickorish. Heegaard
Heegaard_splitting
Continuous deformation between two continuous functions
version of a homotopy equivalence) Homeotopy Homotopy type theory Mapping class group Poincaré conjecture Regular homotopy "Homotopy Definition & Meaning"
Homotopy
Representation of a modular tensor category
{\displaystyle {\mathcal {C}}} arrises naturally as the representation of the mapping class group of the torus associated to the Reshetikhin–Turaev topological quantum
Modular_group_representation
Topics referred to by the same term
Galaxies (astronomy) Mapping class group (mathematics) MCWG, Meta-Certificate Working Group (previously Meta-Certificate Group) (Internet security) Make
MCG_(disambiguation)
Natural number
84 is the limit superior of the largest finite subgroup of the mapping class group of a genus g {\displaystyle g} surface divided by g {\displaystyle
84_(number)
Mathematical representation
representations. Consider the braid group Bn to be the mapping class group of a disc with n marked points Dn. The homology group H1(Dn) is free abelian of rank
Burau_representation
American mathematician
research fields include geometric group theory and low-dimensional topology, with a particular focus on mapping class groups of surfaces. Margalit earned his
Dan_Margalit_(mathematician)
Mathematical concept
groups and mapping class groups. An even more general notion is that of an acylindrically hyperbolic group. Acylindricity of an action of a group G {\displaystyle
Hyperbolic_group
Outer automorphism group of a free group on n generators
generators of any finitely generated group. Despite geometric analogies with general linear groups and mapping class groups, their complexity is generally regarded
Out(Fn)
Mathematical behavior near singularities
leading to the Riemann existence theorem. Braid group Monodromy matrix Monodromy theorem Mapping class group (of a punctured disk) König, Wolfgang; Sprekels
Monodromy
hyperbolic group and of a relatively hyperbolic group and includes a significantly wider class of examples, such as mapping class groups and Out(Fn)
Acylindrically hyperbolic group
Acylindrically_hyperbolic_group
Mathematician
community. Tillmann, Ulrike (1997). "On the homotopy of the stable mapping class group". Inventiones Mathematicae. 130 (2): 257–275. Bibcode:1997InMat.130
Ulrike_Tillmann
American mathematician
contributions to the study of knots, 3-manifolds, mapping class groups of surfaces, geometric group theory, contact structures and dynamical systems.
Joan_Birman
Mathematical rule
mapping class group of the space minus a periodic orbit. For example, Peter Kloeden showed that Sharkovskii's theorem holds for triangular mappings,
Sharkovskii's_theorem
Theorem in geometric topology
Jakob Nielsen (1932, pp. 147–148) about whether finite subgroups of mapping class groups can act on surfaces, that was answered positively by Steven Kerckhoff (1980
Nielsen_realization_problem
Aspect of group theory in mathematics
studying Schottky-type subgroups of mapping class groups of Riemann surfaces, where the set on which the mapping class group acts is the Thurston boundary of
Ping-pong_lemma
Gives necessary and sufficient conditions for two braids to have equivalent closures
mathematician Joan Birman published a monograph, Braids, Links, and Mapping Class Groups, based on a graduate course she taught as a visiting professor at
Markov_theorem
Group of symmetries of an n-dimensional hypercube
braid group of n strands in an annular region. The Artin–Tits group of S n ± {\displaystyle S_{n}^{\pm }} is also isomorphic to the mapping class group of
Hyperoctahedral_group
American mathematician
Fellow in 2014. Behrstock, Jason A. "Asymptotic geometry of the mapping class group and Teichmüller space". Geom. Topol. 10 (2006), pages 1523–1578.
Jason_Behrstock
Group type in algebra
graphs Crystallographic groups Molecular symmetry groups Mapping class groups appear in topological quantum field theories Knot groups are used to study molecular
Finitely_generated_group
Type of mathematical theorem
linear groups" (PDF). Invent. Math. 60: 269–295. doi:10.1007/bf01390018. Harer, J. L. (1985). "Stability of the homology of the mapping class groups of orientable
Homological_stability
Roman surface Steiner surface Alexander horned sphere Klein bottle Mapping class group Dehn twist Nielsen–Thurston classification Moise's Theorem (see also
List of geometric topology topics
List_of_geometric_topology_topics
Mathematician
interests concern low-dimensional topology and geometric group theory, including mapping class groups and Teichmüller theory. Tao was a student of Howard Masur
Jing_Tao
American mathematician (born 1967)
Mathematical Exposition. Benson Farb; Dan Margalit (2012). A Primer on Mapping Class Groups. Princeton University Press. ISBN 978-0-691-14794-9. MR 2850125.
Benson_Farb
type of the bundle obtained depends only on the conjugacy class, in the mapping class group, of the gluing homeomorphism chosen. This construction is
Surface bundle over the circle
Surface_bundle_over_the_circle
In mathematics, relatively hyperbolic groups form an important class of groups of interest for geometric group theory. The main purpose in their study
Relatively_hyperbolic_group
Mathematical theorem in group theory
groups satisfying the Tits alternative which are either not linear, or at least not known to be linear, are: Hyperbolic groups Mapping class groups;
Tits_alternative
Russian mathematician (born 1954)
classification of subgroups of surface mapping class groups, and the establishment that surface mapping class groups satisfy the Tits alternative. He is
Nikolai Ivanov (mathematician)
Nikolai_Ivanov_(mathematician)
Two-dimensional manifold
{\displaystyle \Sigma _{g,k},} for example in the study of the mapping class group. Non-compact surfaces are more difficult to classify. As a simple
Surface_(topology)
Relation between Dehn twists
relation is a relation that appears between certain Dehn twists in the mapping class group of a surface. The most general version of the relation involves seven
Lantern_relation
Subfield of mathematical topology
3-manifold, given input a word (in Dehn twist generators) for the mapping class group of a surface. The 3-manifold is the one that uses the word as the
Computational_topology
American mathematician
geometric group theory, which involves the intersection of algebra and low-dimensional topology. In particular, she studies mapping class group of surfaces
Tara_E._Brendle
American mathematician
Smale. Allen Hatcher and William Thurston, A presentation for the mapping class group of a closed orientable surface, Topology 19 (1980), no. 3, 221–237
Allen_Hatcher
Area in mathematics devoted to the study of finitely generated groups
Free-by-cyclic groups One-relator groups Hyperbolic groups Mapping class groups (automorphisms of surfaces) Symmetric groups Braid groups Coxeter groups General
Geometric_group_theory
American mathematician and novelist
Lochak, P. (1997), 2. The Inverse Galois Problem, Moduli Spaces and Mapping Class Groups, London Mathematical Society Lecture Note Series, vol. 242–243, Cambridge;
Leila_Schneps
Austrian mathematician and physicist
connections, higher categories, topological quantum field theories, the mapping class group, and topological models in condensed matter physics. Earlier work
Catherine_Meusburger
Polygon associated with a compact Riemann surface
homeomorphisms and the fundamental group: this reflects the fact that the mapping class group of a Riemann surface—the group of quasiconformal self-homomorphisms
Fundamental_polygon
Alternative mathematical ordering
to the mapping class group: once-punctured surfaces", in Baumslag, Gilbert (ed.), Geometric and computational perspectives on infinite groups, DIMACS
Cyclic_order
Basic question in geometry and topology
1-dimensional: homeomorphisms of the circle 2-dimensional: mapping class group and Torelli group Analogously to the classification of manifolds, in high
Classification_of_manifolds
American mathematician
symplectic structure is invariant under the natural action of the mapping class group, and using the relationship between Dehn twists and the generalized
William Goldman (mathematician)
William_Goldman_(mathematician)
Set of mathematical functions concerning algebraic group isomorphism
for mapping class groups of closed surfaces. Nielsen transformations were introduced in (Nielsen 1921) to prove that every subgroup of a free group is
Nielsen_transformation
Partitioned topological space
These are used in Thurston's classification of elements of the mapping class group and in his theory of earthquake maps. Quadratic laminations, which
Lamination_(topology)
German mathematician (born 1961)
Mathematical Society in 2017. Hamenstädt, Ursula (2008). "Geometry of the mapping class groups I: Boundary amenability". Inventiones Mathematicae. 175 (3): 545–609
Ursula_Hamenstädt
Concept in mathematics
(2001), "Configuration spaces and braid groups on graphs in robotics", Knots, braids, and mapping class groups—papers dedicated to Joan S. Birman, AMS/IP
Configuration space (mathematics)
Configuration_space_(mathematics)
Tools for studying groups based on techniques from algebraic topology
various other groups such as symmetric groups or mapping class groups. In quantum mechanics we often have systems with a symmetry group G . {\displaystyle
Group_cohomology
(and in particular, its mapping class group); for the case of simple poles, this amounts to the study of the action of braid groups. For the particularly
Isomonodromic_deformation
Mathematics concept
ISSN 0040-9383. MR 0744850. Johannson, Klaus (1979). "On the mapping class group of simple 3-manifolds". In Fenn, Roger A. (ed.). Topology of low-dimensional
Haken_manifold
American mathematician
solved a 50 year old problem posed by Max Dehn on the action of the mapping class group on curves and arcs in surfaces, developed combinatorial aspects of
Robert_Penner
Concept in mathematics
discovered for the groups acting on them. The hyperbolicity of the curve complex has led to new results on the mapping class group. Similarly, the hyperbolicity
Hyperbolic_metric_space
curvature then G is co-Hopfian. The mapping class group of a closed hyperbolic surface is co-Hopfian. The group Out(Fn) (where n>2) is co-Hopfian. Delzant
Co-Hopfian_group
Algebra of formal sums
"Automorphism groups of free groups, surface groups and free abelian groups", in Farb, Benson (ed.), Problems on mapping class groups and related topics
Free_abelian_group
Awarded every year by the American Mathematical Society
Retrieved 2025-11-20. Farb, Benson; Margalit, Dan (2011). A Primer on Mapping Class Groups. Princeton Mathematical Series. Princeton University Press. ISBN 978-0-69114794-9
Leroy_P._Steele_Prize
Branch of mathematics
problems). Other group-theoretic topics like mapping class groups, property (T), solvability, amenability and lattices in Lie groups are sometimes regarded
Geometry
American mathematician (born 1937)
theorem Haboush's theorem Hilbert–Mumford criterion Stable mapping class group Mumford-Tate group Mumford measure Mumford vanishing theorem Manin–Mumford
David_Mumford
Birman published the book Braids, Links, and Mapping Class Groups, which was the first book devoted to braid groups. 1975: American mathematician Julia Robinson
Timeline of women in mathematics
Timeline_of_women_in_mathematics
Algebraic structure
Problems on mapping class groups and related topics. Amer. Math. Soc. p. 357. ISBN 978-0-8218-3838-9. Auslander, M.; Buchsbaum, D. A. (1974). Groups, rings
Semigroup
groups, II: The ending lamination conjecture", Annals of Mathematics, 176 (2012), 1–149. with Jason Behrstock: "Dimension and rank for mapping class groups"
Yair_Minsky
British mathematician
hyperbolic groups. Much of Bowditch's work in 2000s concerns the study of the curve complex, with various applications to 3-manifolds, mapping class groups and
Brian_Bowditch
Diagram to visually organize information
concept mapping concluded that concept mapping is more effective than "reading text passages, attending lectures, and participating in class discussions"
Mind_map
Homotopic map of a graph
an automorphism α of Fn the mapping torus group of α is word-hyperbolic if and only if α has no periodic conjugacy classes; a theorem of Bridson and Groves
Train_track_map
American mathematician and composer (1938–2018)
He is the namesake of the Johnson homomorphism in the study of mapping class groups of surfaces. Johnson’s early talent for mathematics earned him a
Dennis_Johnson_(composer)
Concept in mathematics
Out(Fn) counterpart of the notion of a pseudo-Anosov element of the mapping class group of a finite type surface. Fully irreducibles play an important role
Fully irreducible automorphism
Fully_irreducible_automorphism
Rational function of the form (az + b)/(cz + d)
fractional transformation Liouville's theorem (conformal mappings) Lorentz group Modular group Poincaré half-plane model Projective geometry Projective
Möbius_transformation
Mapping by communities to contest state maps
Counter-mapping is the creation of maps that challenge "dominant power structures, to further seemingly progressive goals". Counter-mapping is used in
Counter-mapping
Type of homotopy group of a topological space
{Homeo}}(X))=MCG^{*}(X)} is the mapping class group for X . {\displaystyle X.} In other words, the mapping class group is the set of connected components
Homeotopy
Danish mathematician
Weiss) proving the Mumford conjecture on the cohomology of the stable mapping class group, and for developing topological cyclic homology theory. Madsen earned
Ib_Madsen
Mathematical function that preserves angles
include orientation-reversing mappings whose Jacobians can be written as any scalar times any orthogonal matrix. For mappings in two dimensions, the
Conformal_map
Class of diffeomorphism
as the mapping class group. It is known (for compact, orientable S) that this is isomorphic with the automorphism group of the fundamental group of S.
Large_diffeomorphism
Element of algebraic structure
Bibcode:1969QJMat..20..235G, doi:10.1093/qmath/20.1.235 Benson Farb, Problems on mapping class groups and related topics (Volume 74 of Proceedings of symposia in pure
Garside_element
American mathematician
2004.8.1471, MR 2119302, S2CID 3131626 Geometric group theory Teichmüller space Mapping class group Train track map Biographies of Candidates 2002. Archived
Karen_Vogtmann
German-American mathematician (1878–1952)
Chiral knot Conjugacy problem Freiheitssatz Group isomorphism problem Lotschnittaxiom Mapping class group of a surface Non-Archimedean ordered field Scissors
Max_Dehn
Class of Turkish aircraft carriers under construction
The MUGEM-class aircraft carrier is an initiative by the Turkish Navy to build a fully indigenous aircraft carrier. MUGEM is a Turkish acronym for Milli
MUGEM-class_aircraft_carrier
Topological construction
In mathematics, specifically algebraic topology, the mapping cylinder of a continuous function f {\displaystyle f} between topological spaces X {\displaystyle
Mapping_cylinder
prescribed regularity. Manifold with boundary Manifold with corners Mapping class group Morse function Neat submanifold – A submanifold whose boundary equals
Glossary of differential geometry and topology
Glossary_of_differential_geometry_and_topology
Group theory
bijective residue-class-wise affine mappings. A mapping f : Z → Z {\displaystyle f:\mathbb {Z} \rightarrow \mathbb {Z} } is called residue-class-wise affine
Residue-class-wise affine group
Residue-class-wise_affine_group
Graph drawing used to study Riemann surfaces
Galois actions II. The inverse Galois problem, moduli spaces and mapping class groups. Proceedings of the conference on geometry and arithmetic of moduli
Dessin_d'enfant
List of questions aimed at obtaining data from a group of people
list of questions aimed for extracting specific data from a particular group of people. Surveys may be conducted by phone, mail, via the internet, and
Survey_(human_research)
In number theory, measure of non-unique factorization
the task. The mapping from rings of integers R {\displaystyle R} to their corresponding class groups is functorial, and the class group can be subsumed
Ideal_class_group
Graduate-level textbooks in mathematics
82 Braids, Links, and Mapping Class Groups. Joan S. Birman 1975-02-01 237 978-0691081496 83 Automorphic Forms on Adele Groups. Stephen S. Gelbart 1975-03-21
Annals_of_Mathematics_Studies
groups and surface groups by convex cocompact subgroups of Out t ( F n ) {\displaystyle \operatorname {Out} t(F_{n})} and of mapping class groups.
Cannon–Thurston_map
MAPPING CLASS-GROUP
MAPPING CLASS-GROUP
Surname or Lastname
English (common in Lancashire and northern Ireland)
English (common in Lancashire and northern Ireland) : from a patronymic or pet form of Topp, or possibly from an unattested Old English personal name Topping.
Surname or Lastname
English
English : nickname from Old French, Middle English cras ‘big’, ‘fat’ (Latin crassus).Possibly an altered spelling of German Krass.
Girl/Female
Muslim
Glass
Boy/Male
Australian, Dutch, German, Greek
People's Victory
Surname or Lastname
English
English : patronymic from Mann 1 and 2.Irish : adopted as an English equivalent of Gaelic Ó MainnÃn ‘descendant of MainnÃn’, probably an assimilated form of MainchÃn, a diminutive of manach ‘monk’. This is the name of a chieftain family in Connacht. It is sometimes pronounced Ó MaingÃn and Anglicized as Mangan.Anstice Manning, widow of Richard Manning of Dartmouth, England, came to MA with her children in 1679. Her great-great-grandson Robert, born at Salem, MA, in 1784, was the uncle and protector of author Nathaniel Hawthorne. Another early bearer of the relatively common British name was Jeffrey Manning, one of the earliest settlers in Piscataway township, Middlesex Co., NJ. His great-grandson James Manning (1738–91) was a founder and the first president of Rhode Island College (Brown University).
Surname or Lastname
English
English : from the medieval female personal name Cass, a short form of Cassandra. This was the name (of uncertain, possibly non-Greek, origin) of an ill-fated Trojan prophetess of classical legend, condemned to foretell the future but never be believed; her story was well known and widely popular in medieval England.
Girl/Female
Indian
Glass
Surname or Lastname
English
English : variant of Close 1.German : variant of Kloss.
Surname or Lastname
English (Devon)
English (Devon) : variant spelling of Appling.
Male
German
Short form of German Niclaus, CLAUS means "victor of the people."Â
Girl/Female
Indian
Glass
Female
English
English short form of Latin Cassandra, CASS means "she who entangles men."Â
Surname or Lastname
English and German
English and German : metonymic occupational name for a glazier or glass blower, from Old English glæs ‘glass’ (akin to Glad, referring originally to the bright shine of the material), Middle High German glas.Irish and Scottish : Anglicized form of the epithet glas ‘gray’, ‘green’, ‘blue’ or any of various Gaelic surnames derived from it.German : altered form of the personal name Klass, a reduced form of Nikolaus (see Nicholas).Jewish (Ashkenazic) : ornamental name from German Glass ‘glass’, or a metonymic occupational name for a glazier or glass blower.
Surname or Lastname
English
English : from the medieval personal name Classe, a short form of Nicholas. See also Clayson.Variant of Klaas or Klass, North German forms of Claus.
Surname or Lastname
North German
North German : topographic name from Middle Low German plas ‘place’, ‘open square’, ‘street’.South German (also Pläss) : from a short form of the medieval personal name Blasius.English : variant of Place 3.
Girl/Female
Tamil
Glass
Surname or Lastname
English
English : from Old English Tæpping, an unattested patronymic from Tæppa. Compare Tapp.Joseph Tapping (d. 1678) is buried in King’s Chapel Burying Ground, Boston, MA.
Surname or Lastname
English and Irish
English and Irish : probably a hypercorrected form of Lappin.
Boy/Male
Australian, Danish, Dutch, Greek, Swedish
People of Victory; Victory of the People
Boy/Male
Greek Latin
People's victory.
MAPPING CLASS-GROUP
MAPPING CLASS-GROUP
Surname or Lastname
English
English : see Truesdale.
Girl/Female
Bengali, Gujarati, Hindu, Indian, Kannada, Malayalam, Marathi, Sindhi, Tamil, Telugu
A Star
Girl/Female
Biblical
God's redemption.
Biblical
the grace or mercy of the Lord,Jehovah's gift: the same name as Johanan, a contraction of Jehohanan
Male
Hebrew
(×–Ö°×‘Ö´×™× Ö¸×”) Hebrew name ZEBIYNA means "bought." In the bible, this is the name of a son of Nebo who took a foreign wife.
Male
German
 Possibly a variant spelling of German Armin, ARMEN means "army man." Compare with another form of Armen.
Boy/Male
Australian, Scottish
River
Girl/Female
Arabic, Muslim, Pashtun
Nomad
Boy/Male
Hindu, Indian
The Ocean
Boy/Male
Gujarati, Hindu, Indian
Powerful
MAPPING CLASS-GROUP
MAPPING CLASS-GROUP
MAPPING CLASS-GROUP
MAPPING CLASS-GROUP
MAPPING CLASS-GROUP
v. t.
To smooth or polish anything, as leater, by rubbing it with a glass burnisher.
n.
A small sandglass, running about three minutes, for marking time in boiling eggs; also, a small glass for holding an egg, at table.
v. t.
Variant of Clasp
v. t.
A looking-glass; a mirror.
v. t.
To case in glass.
n.
One of the sections into which a church or congregation is divided, and which is under the supervision of a class leader.
v. t.
Anything made of glass.
n.
To arrange in classes; to classify or refer to some class; as, to class words or passages.
a.
Pertaining to the harp; as, harping symphonies.
v. t.
To cover or furnish with glass; to glaze.
a.
Of the best class; of the highest rank; in the first division; of the best quality; first-rate; as, a first-class telescope.
a.
Of the rank or degree below the best highest; inferior; second-rate; as, a second-class house; a second-class passage.
n.
A cupping glass.
n.
A kind of machine blanket or wrapping material used by calico printers.
v. t.
To shut or fasten together with, or as with, a clasp; to shut or fasten (a clasp, or that which fastens with a clasp).
n.
To divide into classes, as students; to form into, or place in, a class or classes.
n.
The process of making, or of becoming malt.
n.
A group of individuals ranked together as possessing common characteristics; as, the different classes of society; the educated class; the lower classes.
a.
Biting; pinching; painful; destructive; as, a nipping frost; a nipping wind.