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TOPOLOGY
 Topology is a math discipline who treats of the research of invariants
in a geometry cleared of all idea of measurement or of distance. Applied to the learning of surfaces, for example, it studies the invariants who resist to the distortion of these.
Of recent apparition (middle of XIXème, Listing, Möbius, etc..), topology abandons all metrics in order to interest of qualitative manner to the spatial relationships between the different parts of figures. Lacan shows that the logic of the unconscious refers to the topology. Starting from 1961, he clings to show the nature topologic of the joint of the principal concepts of the psychoanalysis. Some patterns like the band of Möbius, the torus, the cross-cap, the bottle of Klein or again different types of knots realize respectively for Lacan the constitution of the subject, of some dealings between the ask and the desire, of fantasies, of the nature of the object a ,[etc]. But for the psychoanalysis, topology is not only a pattern: Lacan learn us to consider it like the very structure.
Here are fundamental objects of the lacanian topology :
 The Möbius band
 The torus
 The Klein bottle
 The cross-cap
 The borromean knot
Topology is a branch of pure mathematics, deals with the fundamental
properties of abstract spaces. Whereas classical geometry is concerned
with measurable quantities, such as angle, distance, area, and so forth,
topology is concerned with notations of continuity and relative position.
Point-set topology regards geometrical figures as collections of points,
with the entire collection often considered a space. Combinatorial or algebraic
topology treats geometrical figures as aggregates of smaller building blocks.
BASIC CONCEPTS
In general, topologists study properties of spaces that remain unchanged,
no matter how the spaces are bent, stretched, shrunk, or twisted. Such
transformations of ideally elastic objects are subject only to the condition
that nearby points in one space correspond to nearby points in transformed
version of that space. Because allowed deformation can be carried out by
manipulating a rubber sheet, topology is sometimes known as rubber-sheet
geometry. In contrast, cutting, then gluing together parts of a space is
bound to fuse two or more points and to separate points once close together.
The basic ideas of topology surfaced in the mid-19th century as offshoots
of algebra and ANALYTIC GEOMETRY. Now the field is a major mathematical
pursuit, with applications ranging from cosmology and particle physics
to the geometrical structure of proteins and other molecules of biological
interest.
In essence, the topology of a space provides a way of telling which
points are very close to one another and which are not. For instance, it
supplies a way of determining whether a curve drawn on a space is continuous
(unbroken) or not. However, it does not distinguish between a smooth and
crinkly curve or say anything about the curve's length. Topology does not
deal with smoothness or size. Hence, a cube's surface has the same topology
as a sphere's surface, even though one has sharp corners and the other
smooth. Such topologically identical figures are termed homeomorphic. On
the other hand, the surface of a sphere has a difficult topology from the
surface of a torus (doughnut-shaped space).
A primary aim of topology is to find a serviceable set of rules or
procedures for recognizing spaces in all dimensions. In such a classification
scheme, two spaces would belong to the same topological class if they had
the same basic, overall structure although they might differ drastically
in their details.
The simplest topological spaces are known as Euclidean spaces. In general,
the term DIMENSION signifies an independent parameter, or coordinate. A
space has n dimensions if each of its points is completely determined by
n independent numbers. An infinitely long line is a one-dimension Euclidean
space. The plane is a two-dimensional Euclidean space. The space of ordinary
experience is usually considered a three-dimensional Euclidean space.
MANIFOLDS
The term manifold covers more complicated types than Euclidean spaces.
Manifolds locally appear flat, or Euclidean, but on a larger scale may
bend and twist into exotic and intricate forms. Any surface, however curved
and complicated, so long as it does not intersect itself, can be thought
of as consisting of small, two-dimensional, Euclidean patches glued together.
Special manifold characteristics, often expressed as numbers or algebraic
expressions, help distinguish manifolds. Such expressions, known as topological
invariants, provide a convenient way of categorizing manifolds. Dimension,
the number of coordinates required to specify a point in a given space,
is one example of a topological invariant. Manifolds may also be either
bounded or unbounded. A circle is an example of a bounded, one-dimensional
manifold, but a line stretching off indefinitely in both directions is
unbounded. The same distinction applies to spaces of any dimension. Certain
types of two-manifolds, termed compact surfaces, can be classified in terms
of an invariant called the Euler number. For a compact surface divided
into triangles, the surface's Euler number equals the number of triangles
minus the number of edges plus the number of vertices (a vertex is the
point of a triangle farthest from its base). No matter how the surface
is divided up into triangles--so long as no triangle's vertex rests on
another triangle's edge--each different type of compact surface has a particular
Euler number. The sphere has Euler number 2; the torus has Euler number
0; the surface of a two-handled soup tureen has a Euler number of -2. Each
even integer less than or equal to 2 is the Euler number for exactly one
type of closed surface.
The same idea can be expressed in terms of spheres to which are attached
a certain number of handles. The surface of a sphere or a lump of clay
fall into one group, whereas the surface of a doughnut or a coffee mug
fall into another. Because both forms have one hole, one can imagine smoothly
deforming a doughnut-shaped piece of clay to produce a mug with a single
handle. On the other hand, there is no way, short of cutting, to turn a
spherical balloon into an inner tube.
Topologists, using suitable invariations, can examine in detail what
manifolds look like and how one can be transformed into another. Much manifold
study concerns the search for more finely tuned invariants that make subtler
distinctions. Because manifolds in higher dimensions are impossible to
visualize, these invariants often stand in for the manifolds.
MANIFOLD CLASSIFICATION
Mathematicians have developed workable schemes for studying manifolds
in every dimension except three and four. Dimension three remains a puzzle
because proposed classification schemes cover only a portion of all conceivable
three-manifolds. Recent attempts to demystify four-dimensional spaces reveal
them to have special characteristics quite unlike those of any dimension,
higher or lower. The Poincare conjecture, named for the French mathematician
Henri POINCARE, ranks as one of the most baffling and challenging unsolved
problems in algebraic topology. The central idea of homotopy theory (the
theory of the relationship between topologically identical spaces) is to
reduce topological questions to abstract algebra by associating with topological
spaces various algebraic invariants. Poincare's contribution was the invention
of an abstract concept called the fundamental group for distinguishing
different categories of two-dimensional surfaces. Poincare was able to
show that any two-dimensional surface having the same fundamental group
as the two-dimensional surface of a sphere is topologically equivalent
to such a sphere. He then conjectured that the same relationship holds
for three-dimensional manifolds, and other mathematicians extended the
idea to higher dimensions. Ironically, mathematicians have provided the
equivalent of Poincare's conjecture for all dimensions except three. The
central problem in proving the conjecture in three dimensions is that,
unlike the two-manifolds cases, topologists have no complete classification
scheme for three-manifolds. There exists no list of all possible manifolds
that can be checked one by one to make sure that all have different homotopy
groups.
KNOT THEORY
Topology is also concerned with the ways in which one manifold may
be embedded within another, such as the ways a knotted circle may be embedded
in three-dimensional space. A mathematical knot is the abstract result
of first looping and interlacing a piece of string, then joining its ends
together. Because any such knot is always topologically equivalent to a
circle, the central questions in knot theory concern how that curve is
embedded in three-dimensional space.
Knot theorists are particularly interested in identifying when curves
are truly knotted, in finding ways of distinguishing different knots, and,
more generally, in classifying all possible knots. Although a competent
scout or sailor can readily identify and distinguish between a reef knot
and a granny knot, mathematicians have a tougher task because they must
deal with all conceivable knots. In many cases, two knots may look the
same when, in fact, they are different. Alternatively, a knot may be so
contorted that its true identity is masked.
To make knot classification easier, investigators examine the two-dimensional
shadows cast by the three-dimensional knots. Even the most tangled configuration
can be pictured as a continuous loop whose shadow winds across a flat surface,
sometimes crossing over, and sometimes crossing under itself. One convenient
measure of a knot's complexity is the minimum number of crossings that
show up after looking at all possible shadows of a particular knot. A loop
without any twists of crossings (in its simplest form, a circle) is called
an unknot. The simplest possible knot is the overhand, or trefoil, knot,
which is really just a circle that winds through itself. In its plainest
form, this knot has three crossings. It also comes in two forms: left-handed
and right-handed configurations, which are mirror images of each other.
Knot theorists have identified 12,965 distinct knots with 13 or fewer crossings.
A more sophisticated approach to distinguishing knots is to use the
arrangement of crossings in a knot diagram to produce an algebraic expression--a
polynomial invariant--that serves as a label for the knot. Recent discoveries
of a wide range of new invariants show promise because they seem to distinguish
more different types of knots than previously known invariants.
Ivars Peterson
Bibliography:
Francis, G. K., A Topological Picturebook (1987); Firby, P, A., and
Gardiner, C. F., Surface Topology (1982); Barr, Stephen, Experiments in
Topology (1972); Weeks, Jeffrey R., The Shape of Space (1985); Peterson,
Ivars, Islands of Truth (1990) and The Mathematical Tourist (1988); Stewart,
Ian, The Problem of Mathematics (1987).
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