n-connected

n-connected

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".

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n-connected space

A topological space 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

\pi_i(X) \equiv 0~, \quad 1\leq i\leq n ,

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:

π0(X, * ): = [(S0, * ),(X, * )];

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

\pi_i(X) \equiv 0, \quad 0\leq i\leq n.

Examples

  • As described above, a space X is 0-connected if and only if it is path-connected.
  • 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<n.

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 f\colon X \to Y is n-connected if and only if:

  • \pi_i(f)\colon \pi_i(X) \overset{\sim}{\to} \pi_i(Y) is an isomorphism for i < n, and
  • \pi_n(f)\colon \pi_n(X) \twoheadrightarrow \pi_n(Y) 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:

\pi_n(X) \overset{\pi_n(f)}{\to} \pi_n(Y) \to \pi_n(Cf).

If the group on the right πn(Cf) 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 A \hookrightarrow X 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 A \hookrightarrow X to be 1-connected, it must be:

  • onto π0(X),
  • one-to-one on \pi_0(A) \to \pi_0(X), and
  • onto π1(X).

One-to-one on \pi_0(A) \to \pi_0(X) means that if there is a path connecting two points a, b \in A by passing through X, there is a path in A connecting them, while onto π1(X) 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 \pi_{n-1}(A) \to \pi_{n-1}(X) only implies that any element of πn 1(A) that are homotopic in X are abstractly homotopic in Athe homotopy in A may be unrelated to the homotopy in Xwhile being n-connected (so also onto πn(X)) 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-spheremeans 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 theorem 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 M \to N, into a more general topological space, such as the space of all continuous maps between two associated spaces X(M) \to X(N), 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


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