- Source: Homotopical connectivity
In algebraic topology, homotopical connectivity is a property describing a topological space based on the dimension of its holes. In general, low homotopical connectivity indicates that the space has at least one low-dimensional hole. The concept of n-connectedness generalizes the concepts of path-connectedness and simple connectedness.
An equivalent definition of homotopical connectivity is based on the homotopy groups of the space. A space is n-connected (or n-simple connected) if its first n homotopy groups are trivial.
Homotopical connectivity is defined for maps, too. A map is n-connected if it is an isomorphism "up to dimension n, in homotopy".
Definition using holes
All definitions below consider a topological space X.
A hole in X is, informally, a thing that prevents some suitably-placed sphere from continuously shrinking to a point.: 78 Equivalently, it is a sphere that cannot be continuously extended to a ball. Formally,
A d-dimensional sphere in X is a continuous function
f
d
:
S
d
→
X
{\displaystyle f_{d}:S^{d}\to X}
.
A d-dimensional ball in X is a continuous function
g
d
:
B
d
→
X
{\displaystyle g_{d}:B^{d}\to X}
.
A d-dimensional-boundary hole in X is a d-dimensional sphere that is not nullhomotopic (- cannot be shrunk continuously to a point). Equivalently, it is a d-dimensional sphere that cannot be continuously extended to a (d+1)-dimensional ball. It is sometimes called a (d+1)-dimensional hole (d+1 is the dimension of the "missing ball").
X is called n-connected if it contains no holes of boundary-dimension d ≤ n.: 78, Sec.4.3
The homotopical connectivity of X, denoted
conn
π
(
X
)
{\displaystyle {\text{conn}}_{\pi }(X)}
, is the largest integer n for which X is n-connected.
A slightly different definition of connectivity, which makes some computations simpler, is: the smallest integer d such that X contains a d-dimensional hole. This connectivity parameter is denoted by
η
π
(
X
)
{\displaystyle \eta _{\pi }(X)}
, and it differs from the previous parameter by 2, that is,
η
π
(
X
)
:=
conn
π
(
X
)
+
2
{\displaystyle \eta _{\pi }(X):={\text{conn}}_{\pi }(X)+2}
.
= Examples
=A 2-dimensional hole (a hole with a 1-dimensional boundary) is a circle (S1) in X, that cannot be shrunk continuously to a point in X. An example is shown on the figure at the right. The yellow region is the topological space X; it is a pentagon with a triangle removed. The blue circle is a 1-dimensional sphere in X. It cannot be shrunk continuously to a point in X; therefore; X has a 2-dimensional hole. Another example is the punctured plane - the Euclidean plane with a single point removed,
R
2
∖
{
(
0
,
0
)
}
{\displaystyle \mathbb {R} ^{2}\setminus \{(0,0)\}}
. To make a 2-dimensional hole in a 3-dimensional ball, make a tunnel through it. In general, a space contains a 1-dimensional-boundary hole if and only if it is not simply-connected. Hence, simply-connected is equivalent to 1-connected. X is 0-connected but not 1-connected, so
conn
π
(
X
)
=
0
{\displaystyle {\text{conn}}_{\pi }(X)=0}
. The lowest dimension of a hole is 2, so
η
π
(
X
)
=
2
{\displaystyle \eta _{\pi }(X)=2}
.
A 3-dimensional hole (a hole with a 2-dimensional boundary) is shown on the figure at the right. Here, X is a cube (yellow) with a ball removed (white). The 2-dimensional sphere (blue) cannot be continuously shrunk to a single point. X is simply-connected but not 2-connected, so
conn
π
(
X
)
=
1
{\displaystyle {\text{conn}}_{\pi }(X)=1}
. The smallest dimension of a hole is 3, so
η
π
(
X
)
=
3
{\displaystyle \eta _{\pi }(X)=3}
.
For a 1-dimensional hole (a hole with a 0-dimensional boundary) we need to consider
S
0
{\displaystyle S^{0}}
- the zero-dimensional sphere. What is a zero dimensional sphere? - For every integer d, the sphere
S
d
{\displaystyle S^{d}}
is the boundary of the (d+1)-dimensional ball
B
d
+
1
{\displaystyle B^{d+1}}
. So
S
0
{\displaystyle S^{0}}
is the boundary of
B
1
{\displaystyle B^{1}}
, which is the segment [0,1]. Therefore,
S
0
{\displaystyle S^{0}}
is the set of two disjoint points {0, 1}. A zero-dimensional sphere in X is just a set of two points in X. If there is such a set, that cannot be continuously shrunk to a single point in X (or continuously extended to a segment in X), this means that there is no path between the two points, that is, X is not path-connected; see the figure at the right. Hence, path-connected is equivalent to 0-connected. X is not 0-connected, so
conn
π
(
X
)
=
−
1
{\displaystyle {\text{conn}}_{\pi }(X)=-1}
. The lowest dimension of a hole is 1, so
η
π
(
X
)
=
1
{\displaystyle \eta _{\pi }(X)=1}
.
A 0-dimensional hole is a missing 0-dimensional ball. A 0-dimensional ball is a single point; its boundary
S
−
1
{\displaystyle S^{-1}}
is an empty set. Therefore, the existence of a 0-dimensional hole is equivalent to the space being empty. Hence, non-empty is equivalent to (-1)-connected. For an empty space X,
conn
π
(
X
)
=
−
2
{\displaystyle {\text{conn}}_{\pi }(X)=-2}
and
η
π
(
X
)
=
0
{\displaystyle \eta _{\pi }(X)=0}
, which is its smallest possible value.
A ball has no holes of any dimension. Therefore, its connectivity is infinite:
η
π
(
X
)
=
conn
π
(
X
)
=
∞
{\displaystyle \eta _{\pi }(X)={\text{conn}}_{\pi }(X)=\infty }
.
= Homotopical connectivity of spheres
=In general, for every integer d,
conn
π
(
S
d
)
=
d
−
1
{\displaystyle {\text{conn}}_{\pi }(S^{d})=d-1}
(and
η
π
(
S
d
)
=
d
+
1
{\displaystyle \eta _{\pi }(S^{d})=d+1}
): 79, Thm.4.3.2 The proof requires two directions:
Proving that
conn
π
(
S
d
)
<
d
{\displaystyle {\text{conn}}_{\pi }(S^{d})
, that is,
S
d
{\displaystyle S^{d}}
cannot be continuously shrunk to a single point. This can be proved using the Borsuk–Ulam theorem.
Proving that
conn
π
(
S
d
)
≥
d
−
1
{\displaystyle {\text{conn}}_{\pi }(S^{d})\geq d-1}
, that is, that is, every continuous map
S
k
→
S
d
{\displaystyle S^{k}\to S^{d}}
for
k
<
d
{\displaystyle k
can be continuously shrunk to a single point.
Definition using groups
A space X is called n-connected, for n ≥ 0, if it is non-empty, and all its homotopy groups of order d ≤ n are the trivial group:
π
d
(
X
)
≅
0
,
−
1
≤
d
≤
n
,
{\displaystyle \pi _{d}(X)\cong 0,\quad -1\leq d\leq n,}
where
π
i
(
X
)
{\displaystyle \pi _{i}(X)}
denotes the i-th homotopy group and 0 denotes the trivial group. The two definitions are equivalent. The requirement for an n-connected space consists of requirements for all d ≤ n:
The requirement for d=-1 means that X should be nonempty.
The requirement for d=0 means that X should be path-connected.
The requirement for any d ≥ 1 means that X contains no holes of boundary dimension d. That is, every d-dimensional sphere in X is homotopic to a constant map. Therefore, the d-th homotopy group of X is trivial. The opposite is also true: If X has a hole with a d-dimensional boundary, then there is a d-dimensional sphere that is not homotopic to a constant map, so the d-th homotopy group of X is not trivial. In short, X has a hole with a d-dimensional boundary, if-and-only-if
π
d
(
X
)
≇
0
{\displaystyle \pi _{d}(X)\not \cong 0}
.The homotopical connectivity of X is the largest integer n for which X is n-connected.
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 0th homotopy set can be defined as:
π
0
(
X
,
∗
)
:=
[
(
S
0
,
∗
)
,
(
X
,
∗
)
]
.
{\displaystyle \pi _{0}(X,*):=\left[\left(S^{0},*\right),\left(X,*\right)\right].}
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 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
π
i
(
X
)
≃
0
,
0
≤
i
≤
n
.
{\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
:
X
→
Y
{\displaystyle f\colon X\to Y}
is n-connected if and only if:
π
i
(
f
)
:
π
i
(
X
)
→
∼
π
i
(
Y
)
{\displaystyle \pi _{i}(f)\colon \pi _{i}(X)\mathrel {\overset {\sim }{\to }} \pi _{i}(Y)}
is an isomorphism for
i
<
n
{\displaystyle i
, and
π
n
(
f
)
:
π
n
(
X
)
↠
π
n
(
Y
)
{\displaystyle \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
π
n
(
X
)
→
π
n
(
f
)
π
n
(
Y
)
→
π
n
−
1
(
F
f
)
.
{\displaystyle \pi _{n}(X)\mathrel {\overset {\pi _{n}(f)}{\to }} \pi _{n}(Y)\to \pi _{n-1}(Ff).}
If the group on the right
π
n
−
1
(
F
f
)
{\displaystyle \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
↪
X
{\displaystyle 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
↪
X
{\displaystyle 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
↪
X
{\displaystyle A\hookrightarrow X}
to be 1-connected, it must be:
onto
π
0
(
X
)
,
{\displaystyle \pi _{0}(X),}
one-to-one on
π
0
(
A
)
→
π
0
(
X
)
,
{\displaystyle \pi _{0}(A)\to \pi _{0}(X),}
and
onto
π
1
(
X
)
.
{\displaystyle \pi _{1}(X).}
One-to-one on
π
0
(
A
)
→
π
0
(
X
)
{\displaystyle \pi _{0}(A)\to \pi _{0}(X)}
means that if there is a path connecting two points
a
,
b
∈
A
{\displaystyle a,b\in A}
by passing through X, there is a path in A connecting them, while onto
π
1
(
X
)
{\displaystyle \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
π
n
−
1
(
A
)
→
π
n
−
1
(
X
)
{\displaystyle \pi _{n-1}(A)\to \pi _{n-1}(X)}
only implies that any elements of
π
n
−
1
(
A
)
{\displaystyle \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
π
n
(
X
)
{\displaystyle \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.
Lower bounds
Many topological proofs require lower bounds on the homotopical connectivity. There are several "recipes" for proving such lower bounds.
= Homology
=The Hurewicz theorem relates the homotopical connectivity
conn
π
(
X
)
{\displaystyle {\text{conn}}_{\pi }(X)}
to the homological connectivity, denoted by
conn
H
(
X
)
{\displaystyle {\text{conn}}_{H}(X)}
. This is useful for computing homotopical connectivity, since the homological groups can be computed more easily.
Suppose first that X is simply-connected, that is,
conn
π
(
X
)
≥
1
{\displaystyle {\text{conn}}_{\pi }(X)\geq 1}
. Let
n
:=
conn
π
(
X
)
+
1
≥
2
{\displaystyle n:={\text{conn}}_{\pi }(X)+1\geq 2}
; so
π
i
(
X
)
=
0
{\displaystyle \pi _{i}(X)=0}
for all
i
<
n
{\displaystyle i
, and
π
n
(
X
)
≠
0
{\displaystyle \pi _{n}(X)\neq 0}
. Hurewicz theorem: 366, Thm.4.32 says that, in this case,
H
i
~
(
X
)
=
0
{\displaystyle {\tilde {H_{i}}}(X)=0}
for all
i
<
n
{\displaystyle i
, and
H
n
~
(
X
)
{\displaystyle {\tilde {H_{n}}}(X)}
is isomorphic to
π
n
(
X
)
{\displaystyle \pi _{n}(X)}
, so
H
n
~
(
X
)
≠
0
{\displaystyle {\tilde {H_{n}}}(X)\neq 0}
too. Therefore:
conn
H
(
X
)
=
conn
π
(
X
)
.
{\displaystyle {\text{conn}}_{H}(X)={\text{conn}}_{\pi }(X).}
If X is not simply-connected (
conn
π
(
X
)
≤
0
{\displaystyle {\text{conn}}_{\pi }(X)\leq 0}
), then
conn
H
(
X
)
≥
conn
π
(
X
)
{\displaystyle {\text{conn}}_{H}(X)\geq {\text{conn}}_{\pi }(X)}
still holds. When
conn
π
(
X
)
≤
−
1
{\displaystyle {\text{conn}}_{\pi }(X)\leq -1}
this is trivial. When
conn
π
(
X
)
=
0
{\displaystyle {\text{conn}}_{\pi }(X)=0}
(so X is path-connected but not simply-connected), one should prove that
H
0
~
(
X
)
=
0
{\displaystyle {\tilde {H_{0}}}(X)=0}
.
The inequality may be strict: there are spaces in which
conn
π
(
X
)
=
0
{\displaystyle {\text{conn}}_{\pi }(X)=0}
but
conn
H
(
X
)
=
∞
{\displaystyle {\text{conn}}_{H}(X)=\infty }
.
By definition, the k-th homology group of a simplicial complex depends only on the simplices of dimension at most k+1 (see simplicial homology). Therefore, the above theorem implies that a simplicial complex K is k-connected if and only if its (k+1)-dimensional skeleton (the subset of K containing only simplices of dimension at most k+1) is k-connected.: 80, Prop.4.4.2
= Join
=Let K and L be non-empty cell complexes. Their join is commonly denoted by
K
∗
L
{\displaystyle K*L}
. Then:: 81, Prop.4.4.3
conn
π
(
K
∗
L
)
≥
conn
π
(
K
)
+
conn
π
(
L
)
+
2.
{\displaystyle {\text{conn}}_{\pi }(K*L)\geq {\text{conn}}_{\pi }(K)+{\text{conn}}_{\pi }(L)+2.}
The identity is simpler with the eta notation:
η
π
(
K
∗
L
)
≥
η
π
(
K
)
+
η
π
(
L
)
.
{\displaystyle \eta _{\pi }(K*L)\geq \eta _{\pi }(K)+\eta _{\pi }(L).}
As an example, let
K
=
L
=
S
0
=
{\displaystyle K=L=S^{0}=}
a set of two disconnected points. There is a 1-dimensional hole between the points, so the eta is 1. The join
K
∗
L
{\displaystyle K*L}
is a square, which is homeomorphic to a circle, so its eta is 2. The join of this square with a third copy of K is a octahedron, which is homeomorphic to
S
2
{\displaystyle S^{2}}
, and its eta is 3. In general, the join of n copies of
S
0
{\displaystyle S^{0}}
is homeomorphic to
S
n
−
1
{\displaystyle S^{n-1}}
and its eta is n.
The general proof is based on a similar formula for the homological connectivity.
= Nerve
=Let K1,...,Kn be abstract simplicial complexes, and denote their union by K.
Denote the nerve complex of {K1, ... , Kn} (the abstract complex recording the intersection pattern of the Ki) by N.
If, for each nonempty
J
⊂
I
{\displaystyle J\subset I}
, the intersection
⋂
i
∈
J
U
i
{\textstyle \bigcap _{i\in J}U_{i}}
is either empty or (k−|J|+1)-connected, then for every j ≤ k, the j-th homotopy group of N is isomorphic to the j-th homotopy group of K.
In particular, N is k-connected if-and-only-if K is k-connected.: Thm.6
Homotopy principle
In geometric topology, cases when the inclusion of a geometrically-defined space, such as the space of immersions
M
→
N
,
{\displaystyle M\to N,}
into a more general topological space, such as the space of all continuous maps between two associated spaces
X
(
M
)
→
X
(
N
)
,
{\displaystyle 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