Large diffeomorphism
Encyclopedia
In mathematics
and theoretical physics
, a large diffeomorphism is a diffeomorphism
that cannot be continuously connected to the identity diffeomorphism (because it is topologically
non-trivial).
For example, a two-dimensional real torus
has a SL(2,Z)
group of large diffeomorphisms by which the one-cycles
of the torus are transformed into their integer linear combinations. This group of large diffeomorphisms is called the modular group
.
More generally, for a surface
S, the structure of self-homeomorphism
s up to homotopy
is known 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. This is consistent with the genus
1 case, stated above, if one takes into account that then the fundamental group is Z2, on which the modular group acts as automorphisms (as a subgroup of index
2 in all automorphisms, since the orientation may also be reverse, by a transformation with determinant −1).
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
and theoretical physics
Theoretical physics
Theoretical physics is a branch of physics which employs mathematical models and abstractions of physics to rationalize, explain and predict natural phenomena...
, a large diffeomorphism is a diffeomorphism
Diffeomorphism
In mathematics, a diffeomorphism is an isomorphism in the category of smooth manifolds. It is an invertible function that maps one differentiable manifold to another, such that both the function and its inverse are smooth.- Definition :...
that cannot be continuously connected to the identity diffeomorphism (because it is topologically
Topology
Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...
non-trivial).
For example, a two-dimensional real torus
Torus
In geometry, a torus is a surface of revolution generated by revolving a circle in three dimensional space about an axis coplanar with the circle...
has a SL(2,Z)
Special linear group
In mathematics, the special linear group of degree n over a field F is the set of n×n matrices with determinant 1, with the group operations of ordinary matrix multiplication and matrix inversion....
group of large diffeomorphisms by which the one-cycles
of the torus are transformed into their integer linear combinations. This group of large diffeomorphisms is called the modular groupModular group
In mathematics, the modular group Γ is a fundamental object of study in number theory, geometry, algebra, and many other areas of advanced mathematics...
.
More generally, for a surface
Surface
In mathematics, specifically in topology, a surface is a two-dimensional topological manifold. The most familiar examples are those that arise as the boundaries of solid objects in ordinary three-dimensional Euclidean space R3 — for example, the surface of a ball...
S, the structure of self-homeomorphism
Homeomorphism
In the mathematical field of topology, a homeomorphism or topological isomorphism or bicontinuous function is a continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphisms in the category of topological spaces—that is, they are...
s up to homotopy
Homotopy
In topology, two continuous functions from one topological space to another are called homotopic if one can be "continuously deformed" into the other, such a deformation being called a homotopy between the two functions...
is known as the mapping class group
Mapping class group
In mathematics, in the sub-field of geometric topology, the mapping class groupis an important algebraic invariant of a topological space. Briefly, the mapping class group is a discrete group of 'symmetries' of the space.-Motivation:...
. It is known (for compact
Compact space
In mathematics, specifically general topology and metric topology, a compact space is an abstract mathematical space whose topology has the compactness property, which has many important implications not valid in general spaces...
, orientable S) that this is isomorphic with the automorphism group of the fundamental group
Fundamental group
In mathematics, more specifically algebraic topology, the fundamental group is a group associated to any given pointed topological space that provides a way of determining when two paths, starting and ending at a fixed base point, can be continuously deformed into each other...
of S. This is consistent with the genus
Genus (mathematics)
In mathematics, genus has a few different, but closely related, meanings:-Orientable surface:The genus of a connected, orientable surface is an integer representing the maximum number of cuttings along non-intersecting closed simple curves without rendering the resultant manifold disconnected. It...
1 case, stated above, if one takes into account that then the fundamental group is Z2, on which the modular group acts as automorphisms (as a subgroup of index
Index of a subgroup
In mathematics, specifically group theory, the index of a subgroup H in a group G is the "relative size" of H in G: equivalently, the number of "copies" of H that fill up G. For example, if H has index 2 in G, then intuitively "half" of the elements of G lie in H...
2 in all automorphisms, since the orientation may also be reverse, by a transformation with determinant −1).