In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, geometric topology is the study of
manifolds and
maps between them, particularly
embeddings of one manifold into another.
History
Geometric topology as an area distinct from
algebraic topology
Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify ...
may be said to have originated in the 1935 classification of
lens spaces
A lens space is an example of a topological space, considered in mathematics. The term often refers to a specific class of 3-manifolds, but in general can be defined for higher dimensions.
In the 3-manifold case, a lens space can be visualize ...
by
Reidemeister torsion
In mathematics, Reidemeister torsion (or R-torsion, or Reidemeister–Franz torsion) is a topological invariant of manifolds introduced by Kurt Reidemeister for 3-manifolds and generalized to higher dimensions by and .
Analytic torsion (or Ray– ...
, which required distinguishing spaces that are
homotopy equivalent but not
homeomorphic. This was the origin of
''simple'' homotopy theory. The use of the term geometric topology to describe these seems to have originated rather recently.
Differences between low-dimensional and high-dimensional topology
Manifolds differ radically in behavior in high and low dimension.
High-dimensional topology refers to manifolds of dimension 5 and above, or in relative terms, embeddings in
codimension 3 and above.
Low-dimensional topology is concerned with questions in dimensions up to 4, or embeddings in codimension up to 2.
Dimension 4 is special, in that in some respects (topologically), dimension 4 is high-dimensional, while in other respects (differentiably), dimension 4 is low-dimensional; this overlap yields phenomena exceptional to dimension 4, such as
exotic differentiable structures on R4. Thus the topological classification of 4-manifolds is in principle tractable, and the key questions are: does a topological manifold admit a differentiable structure, and if so, how many? Notably, the smooth case of dimension 4 is the last open case of the
generalized Poincaré conjecture; see
Gluck twists.
The distinction is because
surgery theory works in dimension 5 and above (in fact, in many cases, it works topologically in dimension 4, though this is very involved to prove), and thus the behavior of manifolds in dimension 5 and above may be studied using the surgery theory program. In dimension 4 and below (topologically, in dimension 3 and below), surgery theory does not work.
Indeed, one approach to discussing low-dimensional manifolds is to ask "what would surgery theory predict to be true, were it to work?" – and then understand low-dimensional phenomena as deviations from this.
The precise reason for the difference at dimension 5 is because the
Whitney embedding theorem, the key technical trick which underlies surgery theory, requires 2+1 dimensions. Roughly, the Whitney trick allows one to "unknot" knotted spheres – more precisely, remove self-intersections of immersions; it does this via a
homotopy
In topology, a branch of mathematics, two continuous functions from one topological space to another are called homotopic (from grc, ὁμός "same, similar" and "place") if one can be "continuously deformed" into the other, such a deform ...
of a disk – the disk has 2 dimensions, and the homotopy adds 1 more – and thus in codimension greater than 2, this can be done without intersecting itself; hence embeddings in codimension greater than 2 can be understood by surgery. In surgery theory, the key step is in the middle dimension, and thus when the middle dimension has codimension more than 2 (loosely, 2½ is enough, hence total dimension 5 is enough), the Whitney trick works. The key consequence of this is Smale's
''h''-cobordism theorem, which works in dimension 5 and above, and forms the basis for surgery theory.
A modification of the Whitney trick can work in 4 dimensions, and is called
Casson handles – because there are not enough dimensions, a Whitney disk introduces new kinks, which can be resolved by another Whitney disk, leading to a sequence ("tower") of disks. The limit of this tower yields a topological but not differentiable map, hence surgery works topologically but not differentiably in dimension 4.
Important tools in geometric topology
Fundamental group
In all dimensions, the
fundamental group of a manifold is a very important invariant, and determines much of the structure; in dimensions 1, 2 and 3, the possible fundamental groups are restricted, while in dimension 4 and above every
finitely presented group
In mathematics, a presentation is one method of specifying a group. A presentation of a group ''G'' comprises a set ''S'' of generators—so that every element of the group can be written as a product of powers of some of these generators—and ...
is the fundamental group of a manifold (note that it is sufficient to show this for 4- and 5-dimensional manifolds, and then to take products with spheres to get higher ones).
Orientability
A manifold is orientable if it has a consistent choice of
orientation, and a
connected orientable manifold has exactly two different possible orientations. In this setting, various equivalent formulations of orientability can be given, depending on the desired application and level of generality. Formulations applicable to general topological manifolds often employ methods of
homology theory, whereas for
differentiable manifolds more structure is present, allowing a formulation in terms of
differential form
In mathematics, differential forms provide a unified approach to define integrands over curves, surfaces, solids, and higher-dimensional manifolds. The modern notion of differential forms was pioneered by Élie Cartan. It has many application ...
s. An important generalization of the notion of orientability of a space is that of orientability of a family of spaces parameterized by some other space (a
fiber bundle) for which an orientation must be selected in each of the spaces which varies continuously with respect to changes in the parameter values.
Handle decompositions
A
handle decomposition of an ''m''-
manifold ''M'' is a union
:
where each
is obtained from
by the attaching of
-handles. A handle decomposition is to a manifold what a
CW-decomposition is to a topological space—in many regards the purpose of a handle decomposition is to have a language analogous to CW-complexes, but adapted to the world of
smooth manifolds. Thus an ''i''-handle is the smooth analogue of an ''i''-cell. Handle decompositions of manifolds arise naturally via
Morse theory. The modification of handle structures is closely linked to
Cerf theory.
Local flatness
Local flatness is a property of a
submanifold in a
topological manifold of larger
dimension
In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coord ...
. In the
category of topological manifolds, locally flat submanifolds play a role similar to that of
embedded submanifolds in the category of
smooth manifolds.
Suppose a ''d'' dimensional manifold ''N'' is embedded into an ''n'' dimensional manifold ''M'' (where ''d'' < ''n''). If
we say ''N'' is locally flat at ''x'' if there is a neighborhood
of ''x'' such that the
topological pair is
homeomorphic to the pair
, with a standard inclusion of
as a subspace of
. That is, there exists a homeomorphism
such that the
image of
coincides with
.
Schönflies theorems
The generalized
Schoenflies theorem states that, if an (''n'' − 1)-dimensional
sphere
A sphere () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is the c ...
''S'' is embedded into the ''n''-dimensional sphere ''S
n'' in a
locally flat way (that is, the embedding extends to that of a thickened sphere), then the pair (''S
n'', ''S'') is homeomorphic to the pair (''S
n'', ''S''
''n''−1), where ''S''
''n''−1 is the equator of the ''n''-sphere. Brown and Mazur received the
Veblen Prize for their independent proofs
[Mazur, Barry, On embeddings of spheres., ''Bull. Amer. Math. Soc.'' 65 1959 59–65.
] of this theorem.
Branches of geometric topology
Low-dimensional topology
Low-dimensional topology includes:
*
Surfaces (2-manifolds)
*
3-manifolds
*
4-manifolds
each have their own theory, where there are some connections.
Low-dimensional topology is strongly geometric, as reflected in the
uniformization theorem in 2 dimensions – every surface admits a constant curvature metric; geometrically, it has one of 3 possible geometries: positive curvature/spherical, zero curvature/flat, negative curvature/hyperbolic – and the
geometrization conjecture (now theorem) in 3 dimensions – every 3-manifold can be cut into pieces, each of which has one of 8 possible geometries.
2-dimensional topology can be studied as
complex geometry in one variable (Riemann surfaces are complex curves) – by the uniformization theorem every conformal class of metrics is equivalent to a unique complex one, and 4-dimensional topology can be studied from the point of view of complex geometry in two variables (complex surfaces), though not every 4-manifold admits a complex structure.
Knot theory
Knot theory is the study of
mathematical knot
In mathematics, a knot is an embedding of the circle into three-dimensional Euclidean space, (also known as ). Often two knots are considered equivalent if they are ambient isotopic, that is, if there exists a continuous deformation of ...
s. While inspired by knots which appear in daily life in shoelaces and rope, a mathematician's knot differs in that the ends are joined together so that it cannot be undone. In mathematical language, a knot is an
embedding of a
circle
A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is cons ...
in 3-dimensional
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidea ...
, R
3 (since we're using topology, a circle isn't bound to the classical geometric concept, but to all of its
homeomorphism
In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isom ...
s). Two mathematical knots are equivalent if one can be transformed into the other via a deformation of R
3 upon itself (known as an
ambient isotopy); these transformations correspond to manipulations of a knotted string that do not involve cutting the string or passing the string through itself.
To gain further insight, mathematicians have generalized the knot concept in several ways. Knots can be considered in other
three-dimensional spaces and objects other than circles can be used; see ''
knot (mathematics)''. Higher-dimensional knots are
''n''-dimensional spheres in ''m''-dimensional Euclidean space.
High-dimensional geometric topology
In high-dimensional topology,
characteristic classes
In mathematics, a characteristic class is a way of associating to each principal bundle of ''X'' a cohomology class of ''X''. The cohomology class measures the extent the bundle is "twisted" and whether it possesses sections. Characteristic classe ...
are a basic invariant, and
surgery theory is a key theory.
A
characteristic class is a way of associating to each
principal bundle on a
topological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called poin ...
''X'' a
cohomology
In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex. Cohomology can be view ...
class of ''X''. The cohomology class measures the extent to which the bundle is "twisted" — particularly, whether it possesses
sections
Section, Sectioning or Sectioned may refer to:
Arts, entertainment and media
* Section (music), a complete, but not independent, musical idea
* Section (typography), a subdivision, especially of a chapter, in books and documents
** Section sig ...
or not. In other words, characteristic classes are global
invariants which measure the deviation of a local product structure from a global product structure. They are one of the unifying geometric concepts in
algebraic topology
Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify ...
,
differential geometry and
algebraic geometry.
Surgery theory is a collection of techniques used to produce one
manifold from another in a 'controlled' way, introduced by . Surgery refers to cutting out parts of the manifold and replacing it with a part of another manifold, matching up along the cut or boundary. This is closely related to, but not identical with,
handlebody decompositions. It is a major tool in the study and classification of manifolds of dimension greater than 3.
More technically, the idea is to start with a well-understood manifold ''M'' and perform surgery on it to produce a manifold ''M ''′ having some desired property, in such a way that the effects on the
homology,
homotopy groups, or other interesting invariants of the manifold are known.
The classification of
exotic spheres by led to the emergence of surgery theory as a major tool in high-dimensional topology.
See also
*
:Maps of manifolds
*
List of geometric topology topics
This is a list of geometric topology topics, by Wikipedia page. See also:
* topology glossary
* List of topology topics
* List of general topology topics
* List of algebraic topology topics
* Publications in topology
Low-dimensional topology Knot ...
*
Plumbing (mathematics)
References
* R. B. Sher and
R. J. Daverman (2002), ''Handbook of Geometric Topology'', North-Holland. .
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