6Simplicial complexes
II Algebraic Topology
6.1 Simplicial complexes
There are many ways of formulating homology theory, and these are all equivalent
at least for sufficiently sane spaces (such as cell complexes). In this course, we
will be using simplicial homology, which is relatively more intuitive and can be
computed directly. The drawback is that we will have to restrict to a particular
kind of space, known as simplicial complexes. This is not a very serious restriction
per se, since many spaces like spheres are indeed simplicial complexes. However,
the definition of simplicial homology is based on exactly how we view our space
as a simplicial complex, and it will take us quite a lot of work to show that the
simplicial homology is indeed a property of the space itself, and not how we
represent it as a simplicial complex.
We now start by defining simplicial complexes, and developing some general
theory of simplicial complexes that will become useful later on.
Definition (Affine independence). A finite set of points
{a
1
, · · · , a
n
} ⊆ R
m
is
affinely independent iff
n
X
i=1
t
i
a
i
= 0 with
n
X
i=1
t
i
= 0 ⇔ t
i
= 0 for all i.
Example. When n = 3, the following points are affinely independent:
The following are not:
The proper way of understanding this definition is via the following lemma:
Lemma.
a
0
, · · · , a
n
∈ R
m
are affinely independent if and only if
a
1
−a
0
, · · · , a
n
−
a
0
are linearly independent.
Alternatively,
n
+ 1 affinely independent points span an
n
-dimensional thing.
Proof. Suppose a
0
, · · · , a
n
are affinely independent. Suppose
n
X
i=1
λ
i
(a
i
− a
0
) = 0.
Then we can rewrite this as
−
n
X
i=1
λ
i
!
a
0
+ λ
1
a
1
+ · · · + λ
n
a
n
= 0.
Now the sum of the coefficients is 0. So affine independence implies that all
coefficients are 0. So a
1
− a
0
, · · · , a
n
− a
0
are linearly independent.
On the other hand, suppose
a
1
− a
0
, · · · , a
n
− a
0
are linearly independent.
Now suppose
n
X
i=0
t
i
a
i
= 0,
n
X
i=0
t
i
= 0.
Then we can write
t
0
= −
n
X
i=1
t
i
.
Then the first equation reads
0 =
−
n
X
i=1
t
i
!
a
0
+ t
1
a
1
+ · · · + t
n
a
n
=
n
X
i=1
t
i
(a
i
− a
0
).
So linear independence implies all t
i
= 0.
The relevance is that these can be used to define simplices (which are simple,
as opposed to complexes).
Definition (
n
-simplex). An
n
-simplex is the convex hull of (
n
+ 1) affinely
independent points a
0
, · · · , a
n
∈ R
m
, i.e. the set
σ = ha
0
, · · · , a
n
i =
(
n
X
i=0
t
i
a
i
:
n
X
i=0
t
i
= 1, t
i
≥ 0
)
.
The points
a
0
, · · · , a
n
are the vertices, and are said to span
σ
. The (
n
+ 1)-tuple
(t
0
, · · · , t
n
) is called the barycentric coordinates for the point
P
t
i
a
i
.
Example. When n = 0, then our 0-simplex is just a point:
When n = 1, then we get a line:
When n = 2, we get a triangle:
When n = 3, we get a tetrahedron:
The key motivation of this is that simplices are determined by their vertices.
Unlike arbitrary subspaces of
R
n
, they can be specified by a finite amount of
data. We can also easily extract the faces of the simplices.
Definition (Face, boundary and interior). A face of a simplex is a subset (or
subsimplex) spanned by a subset of the vertices. The boundary is the union of
the proper faces, and the interior is the complement of the boundary.
The boundary of
σ
is usually denoted by
∂σ
, while the interior is denoted by
˚σ, and we write τ ≤ σ when τ is a face of σ.
In particular, the interior of a vertex is the vertex itself. Note that these
notions of interior and boundary are distinct from the topological notions of
interior and boundary.
Example. The standard
n
-simplex is spanned by the basis vectors
{
e
0
, · · · ,
e
n
}
in R
n+1
. For example, when n = 2, we get the following:
We will now glue simplices together to build complexes, or simplicial com-
plexes.
Definition. A (geometric) simplicial complex is a finite set
K
of simplices in
R
n
such that
(i) If σ ∈ K and τ is a face of σ, then τ ∈ K.
(ii) If σ, τ ∈ K, then σ ∩ τ is either empty or a face of both σ and τ.
Definition (Vertices). The vertices of
K
are the zero simplices of
K
, denoted
V
K
.
Example. This is a simplicial complex:
These are not:
Technically, a simplicial complex is defined to be a set of simplices, which
are just collections of points. It is not a subspace of
R
n
. Hence we have the
following definition:
Definition (Polyhedron). The polyhedron defined by
K
is the union of the
simplices in K, and denoted by |K|.
We make this distinction because distinct simplicial complexes may have the
same polyhedron, such as the following:
Just as in cell complexes, we can define things like dimensions.
Definition (Dimension and skeleton). The dimension of
K
is the highest
dimension of a simplex of
K
. The
d
-skeleton
K
(d)
of
K
is the union of the
n-simplices in K for n ≤ d.
Note that since these are finite and live inside
R
n
, we know that
|K|
is always
compact and Hausdorff.
Usually, when we are given a space, say
S
n
, it is not defined to be a simplicial
complex. We can “make” it a simplicial complex by a triangulation.
Definition (Triangulation). A triangulation of a space
X
is a homeomorphism
h : |K| → X, where K is some simplicial complex.
Example. Let
σ
be the standard
n
-simplex. The boundary
∂σ
is homeomorphic
to
S
n−1
(e.g. the boundary of a (solid) triangle is the boundary of the triangle,
which is also a circle). This is called the simplicial (n − 1)-sphere.
We can also triangulate our S
n
in a different way:
Example. In
R
n+1
, consider the simplices
h±
e
0
, · · · , ±
e
n
i
for each possible
combination of signs. So we have 2
n+1
simplices in total. Then their union
defines a simplicial complex K, and
|K|
∼
=
S
n
.
The nice thing about this triangulation is that the simplicial complex is
invariant under the antipodal map. So not only can we think of this as a
triangulation of the sphere, but a triangulation of RP
n
as well.
As always, we don’t just look at objects themselves, but also maps between
them.
Definition (Simplicial map). A simplicial map
f
:
K → L
is a function
f
:
V
K
→ V
L
such that if
ha
0
, · · · , a
n
i
is a simplex in
K
, then
{f
(
a
0
)
, · · · , f
(
a
n
)
}
spans a simplex of L.
The nice thing about simplicial maps is that we only have to specify where
the vertices go, and there are only finitely many vertices. So we can completely
specify a simplicial map by writing down a finite amount of information.
It is important to note that we say
{f
(
a
0
)
, · · · , f
(
a
n
)
}
as a set span a simplex
of L. In particular, they are allowed to have repeats.
Example. Suppose we have the standard 2-simplex K as follows:
a
0
a
1
a
2
The following does not define a simplicial map because
ha
1
, a
2
i
is a simplex in
K, but {f (a
1
), f(a
2
)} does not span a simplex:
f(a
0
), f(a
1
) f(a
2
)
On the other hand, the following is a simplicial map, because now
{f
(
a
1
)
, f
(
a
2
)
}
spans a simplex, and note that
{f
(
a
0
)
, f
(
a
1
)
, f
(
a
2
)
}
also spans a 1-simplex
because we are treating the collection of three vertices as a set, and not a
simplex.
f(a
0
), f(a
1
) f(a
2
)
Finally, we can also do the following map:
f(a
0
), f(a
1
) f(a
2
)
The following lemma is obvious, but we will need it later on.
Lemma. If
K
is a simplicial complex, then every point
x ∈ |K|
lies in the
interior of a unique simplex.
As we said, simplicial maps are nice, but they are not exactly what we want.
We want to have maps between spaces.
Lemma. A simplicial map
f
:
K → L
induces a continuous map
|f|
:
|K| → |L|
,
and furthermore, we have
|f ◦ g| = |f| ◦ |g|.
There is an obvious way to define this map. We know how to map vertices,
and then just extend everything linearly.
Proof. For any point in a simplex σ = ha
0
, · · · , a
n
i, we define
|f|
n
X
i=0
t
i
a
i
!
=
n
X
i=0
t
i
f(a
i
).
The result is in
L
because
{f
(
a
i
)
}
spans a simplex. It is not difficult to see this
is well-defined when the point lies on the boundary of a simplex. This is clearly
continuous on σ, and is hence continuous on |K| by the gluing lemma.
The final property is obviously true by definition.