N-connected
Encyclopedia
In the mathematical
branch of algebraic topology
, specifically homotopy theory, n-connectedness is a way to say that a space vanishes or that a map is an isomorphism
"up to dimension n, in homotopy
".
X is said to be n-connected if and only if it is path-connected and its first n homotopy groups vanish identically, that is
where the left-hand side denotes the i-th homotopy group
. The requirement of being path-connected can also be expressed as 0-connectedness, when defining the 0th homotopy set as:
this is only a pointed set
, not a group, unless X is itself a topological group
.
A topological space X is path-connected if and only if its 0th homotopy group vanishes identically, as path-connectedness implies that any two points x1 and x2 in X can be connected with a continuous path which starts in x1 and ends in x2, which is equivalent to the assertion that every mapping
from S0 (a discrete set of two points) to X can be deformed continuously to a constant map. With this definition, we can define X to be n-connected if and only if
A space is 1-connected if and only if it is simply connected. Thus, the term n-connected is a natural generalization of being path-connected or simply connected.
It is obvious from the definition that an n-connected space X is also i-connected for all i.
is n-connected if and only if:
The last condition is frequently confusing; it is because the vanishing of the nth homotopy of the homotopy cofiber Cf corresponds surjection on the nth homotopy groups, in the exact sequence:
If the group on the right
vanishes, then the map on the left is a surjection.
For instance, a simply connected map (1-connected map) is one that is an isomorphism on path-components, and onto the fundamental group.
an n-connected inclusion
is one such that, up to dimension n−1, homotopies in the larger space X can be homotoped into homotopies in the subset A.
For example, for an inclusion map
to be 1-connected, it must be:
One-to-one on
means that if there is a path connecting two points 
by passing through X, there is a path in A connecting them, while onto 
means that in fact a path in X is homotopic to a path in A.
In other words, a function which is an isomorphism on
only implies that any element of 
that are homotopic in X are abstractly homotopic in A – the homotopy in A may be unrelated to the homotopy in X – while being n-connected (so also onto 
) means that (up to dimension n−1) homotopies in X can be pushed into homotopies in A.
This gives a more concrete explanation for the utility of the definition of n-connectedness: for example, a space such that the inclusion of the k-skeleton in n-connected (for n>k) – such as the inclusion of a point in the n-sphere – means that any cells in dimension between k and n are not affecting the homotopy type from the point of view of low dimensions.
which describes the relation between singular homology
and the higher homotopy groups.
In geometric topology
, cases when the inclusion of a geometrically-defined space, such as the space of immersions
into a more general topological space, such as the space of all continuous maps between two associated spaces 
are n-connected are said to satisfy a homotopy principle or "h-principle". There are a number of powerful general techniques for proving h-principles.
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...
branch of algebraic topology
Algebraic topology
Algebraic topology is a branch of mathematics which 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 up to homotopy equivalence.Although algebraic topology...
, specifically homotopy theory, n-connectedness is a way to say that a space vanishes or that a map is an isomorphism
Isomorphism
In abstract algebra, an isomorphism is a mapping between objects that shows a relationship between two properties or operations. If there exists an isomorphism between two structures, the two structures are said to be isomorphic. In a certain sense, isomorphic structures are...
"up to dimension n, in 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...
".
n-connected space
A topological spaceTopological space
Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...
X is said to be n-connected if and only if it is path-connected and its first n homotopy groups vanish identically, that is
where the left-hand side denotes the i-th homotopy group
Homotopy group
In mathematics, homotopy groups are used in algebraic topology to classify topological spaces. The first and simplest homotopy group is the fundamental group, which records information about loops in a space...
. The requirement of being path-connected can also be expressed as 0-connectedness, when defining the 0th homotopy set as:
this is only a pointed set
Pointed set
In mathematics, a pointed set is a set X with a distinguished element x_0\in X, which is called the basepoint. Maps of pointed sets are those functions that map one basepoint to another, i.e. a map f : X \to Y such that f = y_0. This is usually denotedf : \to .Pointed sets may be regarded as a...
, not a group, unless X is itself a topological group
Topological group
In mathematics, a topological group is a group G together with a topology on G such that the group's binary operation and the group's inverse function are continuous functions with respect to the topology. A topological group is a mathematical object with both an algebraic structure and a...
.
A topological space X is path-connected if and only if its 0th homotopy group vanishes identically, as path-connectedness implies that any two points x1 and x2 in X can be connected with a continuous path which starts in x1 and ends in x2, which is equivalent to the assertion that every mapping
Map (mathematics)
In most of mathematics and in some related technical fields, the term mapping, usually shortened to map, is either a synonym for function, or denotes a particular kind of function which is important in that branch, or denotes something conceptually similar to a function.In graph theory, a map is a...
from S0 (a discrete set of two points) to X can be deformed continuously to a constant map. With this definition, we can define X to be n-connected if and only if
Examples
- As described above, a space
It is obvious from the definition that an n-connected space X is also i-connected for all i
n-connected map
The corresponding relative notion to the absolute notion of an n-connected space is an n-connected map, which is almost defined as a map whose homotopy cofiber Cf is an n-connected space. In terms of homotopy groups, it means that a map is n-connected if and only if:
-
is an isomorphism for 
, and -
is a surjection.
The last condition is frequently confusing; it is because the vanishing of the nth homotopy of the homotopy cofiber Cf corresponds surjection on the nth homotopy groups, in the exact sequence:
If the group on the right
vanishes, then the map on the left is a surjection.For instance, a simply connected map (1-connected map) is one that is an isomorphism on path-components, and onto the fundamental group.
Interpretation
This is instructive for a subset:an n-connected inclusion
is one such that, up to dimension n−1, homotopies in the larger space X can be homotoped into homotopies in the subset A.For example, for an inclusion map
to be 1-connected, it must be:
- onto
- one-to-one on
and - onto
One-to-one on
means that if there is a path connecting two points by passing through X, there is a path in A connecting them, while onto means that in fact a path in X is homotopic to a path in A.In other words, a function which is an isomorphism on
only implies that any element of that are homotopic in X are abstractly homotopic in A – the homotopy in A may be unrelated to the homotopy in X – while being n-connected (so also onto ) means that (up to dimension n−1) homotopies in X can be pushed into homotopies in A.This gives a more concrete explanation for the utility of the definition of n-connectedness: for example, a space such that the inclusion of the k-skeleton in n-connected (for n>k) – such as the inclusion of a point in the n-sphere – means that any cells in dimension between k and n are not affecting the homotopy type from the point of view of low dimensions.
Applications
The concept of n-connectedness is used in the Hurewicz theoremHurewicz theorem
In mathematics, the Hurewicz theorem is a basic result of algebraic topology, connecting homotopy theory with homology theory via a map known as the Hurewicz homomorphism...
which describes the relation between singular homology
Singular homology
In algebraic topology, a branch of mathematics, singular homology refers to the study of a certain set of algebraic invariants of a topological space X, the so-called homology groups H_n....
and the higher homotopy groups.
In geometric topology
Geometric topology
In mathematics, geometric topology is the study of manifolds and maps between them, particularly embeddings of one manifold into another.- Topics :...
, cases when the inclusion of a geometrically-defined space, such as the space of immersions
into a more general topological space, such as the space of all continuous maps between two associated spaces are n-connected are said to satisfy a homotopy principle or "h-principle". There are a number of powerful general techniques for proving h-principles.See also
- connected spaceConnected spaceIn topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union of two or more disjoint nonempty open subsets. Connectedness is one of the principal topological properties that is used to distinguish topological spaces...
- simply connected
- path-connected
- homotopy groupHomotopy groupIn mathematics, homotopy groups are used in algebraic topology to classify topological spaces. The first and simplest homotopy group is the fundamental group, which records information about loops in a space...