In the mathematical branch of algebraic topology, specifically homotopy theory, n-connectedness (sometimes, n-simple connectedness) generalizes the concepts of path-connectedness and simple connectedness. To say that a space is n-connected is to say that its first n homotopy groups are trivial, and to say that a map is n-connected means that it is an isomorphism "up to dimension n, in homotopy".
n-connected space
A topological space X is said to be n-connected (for positive n) when it is non-empty, path-connected, and its first n homotopy groups vanish identically, that is
\( {\displaystyle \pi _{i}(X)\cong 0,\quad 1\leq i\leq n,} \)
where \( {\displaystyle \pi _{i}(X)} \) denotes the i-th homotopy group and 0 denotes the trivial group.[1]
The requirements of being non-empty and path-connected can be interpreted as (−1)-connected and 0-connected, respectively, which is useful in defining 0-connected and 1-connected maps, as below. The 0-th homotopy set can be defined as:
\( \pi_0(X,*) := [(S^0,*), (X,*)]. \)
This is only a pointed set, not a group, unless X is itself a topological group; the distinguished point is the class of the trivial map, sending S0 to the base point of X. Using this set, a space is 0-connected if and only if the 0th homotopy set is the one-point set. The definition of homotopy groups and this homotopy set require that X be pointed (have a chosen base point), which cannot be done if X is empty.
A topological space X is path-connected if and only if its 0-th 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
\( {\displaystyle \pi _{i}(X)\simeq 0,\quad 0\leq i\leq n.} \)
Examples
A space X is (−1)-connected if and only if it is non-empty.
A space X is 0-connected if and only if it is non-empty and path-connected.
A space is 1-connected if and only if it is simply connected.
An n-sphere is (n − 1)-connected.
n-connected map
The corresponding relative notion to the absolute notion of an n-connected space is an n-connected map, which is defined as a map whose homotopy fiber Ff is an (n − 1)-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 (n − 1)-st homotopy group of the homotopy fiber Ff corresponds to a surjection on the nth homotopy groups, in the exact sequence:
\( \pi_n(Y) \to \pi_{n-1}(Ff). \)
If the group on the right \( \pi_{n-1}(Ff) \) vanishes, then the map on the left is a surjection.
Low-dimensional examples:
A connected map (0-connected map) is one that is onto path components (0th homotopy group); this corresponds to the homotopy fiber being non-empty.
A simply connected map (1-connected map) is one that is an isomorphism on path components (0th homotopy group) and onto the fundamental group (1st homotopy group).
n-connectivity for spaces can in turn be defined in terms of n-connectivity of maps: a space X with basepoint x0 is an n-connected space if and only if the inclusion of the basepoint \( x_0 \hookrightarrow X \) is an n-connected map. The single point set is contractible, so all its homotopy groups vanish, and thus "isomorphism below n and onto at n" corresponds to the first n homotopy groups of X vanishing.
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 \( \pi_0(X), \)
one-to-one on \( \pi_0(A) \to \pi_0(X) \) , and
onto \( \pi_1(X). \)
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 \( \pi _{1}(X) \) 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 elements of \( \pi_{n-1}(A) \) 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 \( \pi_n(X)) \) 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 where the inclusion of the k-skeleton is n-connected (for n > k) – such as the inclusion of a point in the n-sphere – has the property that any cells in dimensions between k and n do not affect the lower-dimensional homotopy types.
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
Connected space
Connective spectrum
Path-connected
Simply connected
References
"n-connected space in nLab". ncatlab.org. Retrieved 2017-09-18.
Undergraduate Texts in Mathematics
Graduate Studies in Mathematics
Hellenica World - Scientific Library
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