Download Simplicial Objects and Singular Homology

Simplicial Objects and Singular Homology
Helen Gilmour
24th October 2002
The Simplicial Category and Simplicial Objects
The simplicial category ∆ is the category with objects, all finite ordinal numbers n = {0, 1, . . . , n}
and morphisms f : m −→ n all non-decreasing functions (i.e. all functions f for which
f (i) 6 f (j) for i 6 j).
To the simplicial category we can associate the opposite category ∆op . The objects of ∆op
are the objects of ∆, the morphisms of ∆op are the morphisms f op , which are in one-to one
correspondence f 7−→ f op with the morphisms f of ∆. For each morphism f : m −→ n of ∆,
the domain and codomain of the corresponding f op are as in f op : n −→ m, so the direction is
reversed.
If we take a category A, then a simplicial object S in A is defined to be the functor
S : ∆op −→ A i.e. a contavariant functor ∆ −→ A. If A is the category of sets, then the functor
S is known as a simplicial set. Similarly we have simplicial groups and simplicial rings etc.
We have a presentation for the simplicial category; there are morphisms
δin
: n −→ n + 1
σin : n + 1 −→ n
δin {0, . . . , n} = {0, . . . , î, . . . , n + 1}
σin (i) = σin (i + 1)
where δin is an injection and σin a surjection. These morphisms generate the simplicial category
∆ and along with a list of relations, give a presentation of ∆.
The simplicial category ∆, has a direct geometric interpretation by affine simplices which
give a functor
4 : ∆ −→ Top
Singular Homology Groups of a Topological Space
The singular homology of a topological space is a classical example of using simplicial objects.
In order to define the singular homology groups of a topological space X, it is necessary to
introduce some notation and definitions.
We take the n-dimensional Euclidean space n (this is the space of all n-tuples of real
numbers (a1 , a2 , . . . , an )).
The line segment joining two points u, v ∈ n is the set of all points x0 u + x1 v where x0 ,
x1 are real numbers with x0 + x1 = 1, x0 > 0, x1 > 0.
We can say that a subset C of n is convex if it contains the line segment joining any two
of its points. If u0 , . . . , um are m + 1 points of n , the set of all points
R
R
R
u = x0 u0 + . . . + xm um ,
R
xo + x 1 + . . . + x m = 1
xi
>0
is the smallest convex set containing the points u0 , . . . , um and is called the convex hull of
u0 , . . . , um . Here the real numbers xi are the barycentric coordinates of u relative to u0 , . . . , um .
An m-simplex is defined to be the convex hull of m + 1 points, these points are the vertices
of the simplex.
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R
For each dimension n we can take a standard n-simplex 4n in the space n , labelling the
vertices (0, 1, ..., n). This standard n-simplex is the convex hull of the standard basis of n
along with the origin (labelled 0). The standard n-simplex can be expressed by barycentric
coordinates relative to the origin and the standard basis of n .
For any topological space X, a singular n-simplex T in X is just a continuous map
T : 4n −→ X. The word “singular” is used to emphasise that T need not be an embedding.
The free abelian group generated by the singular n-simplices of X is denoted Sn (X) and is
called the singular chain group of X in dimension n.
We can think of the “boundary” of the standard n-simplex 4n as consisting of certain
(n − 1)-dimensional singular simplices which are the “faces” of 4n .
In general 4n has n + 1 faces, with it’s ith face being the linear singular simplex
εi : 4n−1 −→ 4n , i = 0, . . . , n, where εi is a map that carries 4n−1 by a linear homeomorphism
onto the face (0, 1, . . . , î, . . . , n) of 4n , where the vertex i is to be omitted.
So if we have any singular n-simplex T : 4n −→ X, it has n + 1 faces di T defined by the
composite
di T = T εi : 4n−1 −→ X
R
R
so, the ith face omits the ith vertex.
The process of forming iterated faces satisfies the identity
i≤j
di dj T = dj−1 di T
We can now define the boundary homomorphism
∂ : Sn (X) −→ Sn−1 (X)
as the sum of the face homomorphisms di T with alternating signs; that is
n
∂T = d0 T − d1 T + . . . + (−1) dn T =
n
X
(−1)i di T
n>0
i=0
So, ∂T is a formal sum of singular simplices of dimension n − 1 which are the “faces” of T .
We can verify that ∂ 2 = 0, that is that the family of groups Sn (X) and homomorphisms
∂ : Sn (X) −→ Sn−1 (X) for n ≥ 0 forms a chain complex, known as the singular chain complex
of X and denoted S(X).
So we want to prove that the composite ∂∂ : Sn −→ Sn−2 is the zero homomorphism for
n > 1. It is sufficient to prove ∂∂T = 0 for singular n-simplex T : 4n −→ X.
We have
∂∂T
= d0 (d0 T − d1 T + . . . + (−1)n dn T )
−d1 (d0 T − d1 T + . . . + (−1)n dn T )
+(−1)n dn T (d0 T − d1 T + . . . + (−1)n dn T )
X
=
(−1)i+j di dj T
This can be split up as
∂∂T =
X
(−1)i+j di dj T +
j−1<i
X
(−1)i+j di dj T
i≥j
If we use the identity di dj T = dj−1 di T , i ≤ j and switch the labels i and j in the second sum,
we get
X
X
∂∂T =
(−1)i+j dj−1 di T +
(−1)i+k dk di T
j−1≥i
k≥i
The two sums are equal, except for sign and so cancel to give ∂∂ = 0
The homology groups of this chain complex are called the singular homology groups of X
and are denoted Hn (X).
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Computation of Singular Homology Groups
There are several ways of defining homology groups. So far, we have considered singular groups,
however, singular homology theory was introduced as a generalisation of simplicial homology
theory. Simplicial homology groups are easy to understand and compute, however, they are
defined only for particular triangulable spaces. Singular homology groups however are defined
for arbitrary topological spaces and agree with the simplicial homology groups when they are
both defined. It is difficult to compute the singular homology groups of even the simplest of
spaces. The theory of CW-complexes allows singular homology to be computed fairly readily.
As an example, we can compute the homology of a point; a very simple situation, where the
answer is found via a straightforward calculation.
We want to calculate Hn (∗). There is precisely one map 4n −→ ∗. i.e. one singular nsimplex. Therefore, the singular chain group of ∗, Sn (∗) (the free abelian group generated by
n-simplices of ∗) is . If we let αn be the standard generator of the group Sn , then the boundary
homomorphism
∂ : Sn (∗) −→ Sn−1 (∗)
Z
is given by
∂αn =
n
X
i
(−1) αn−1 =
i=0
αn−1 n even
0
n odd
Therefore ∂n = 0 when n is odd and ∂n is an isomorphism when n is even.
For n = 0, we have Hn (∗) = .
Now assume that n > 0. So the singular homology groups of ∗, are the homology groups of
the following singular chain complex;
Z
∂n+1
∂
n
. . . −→ Sn+1 (∗) −→ Sn (∗) −→
Sn−1 (∗) −→ . . .
For n odd, then ∂n = 0 implies that Sn (∗) = ker ∂n . Also ∂n+1 is an isomorphism (since n + 1
is even), hence surjective and so Sn (∗) = im ∂n+1 . Thus
Hn (∗) = ker ∂n im ∂n+1 = 0.
For n > 0 and even, then ∂n is an isomorphism, hence injective and so, ker ∂n = 0. And
Hn (∗) =0 in this case also.
So we have shown that if X is a one point space, then Hn (X) = 0 for all n > 0. This
result is known as the Dimension Axiom for Singular Homology Theory. Homology groups of
all contractible topological spaces coincide with those of a point. In general, if X is a space
with n-components, then H0 (X) is n copies of .
Z
Mayer-Vietoris Sequences
If X is the union of two subspaces X1 and X2 , we can under suitable hypotheses, produce an
exact sequence relating the homology of X with that of X1 and X2 . This exact sequence is
called the Mayer-Vietoris sequence of the pair X1 and X2 .
Suppose we have X = X1 ∪ X2 and that {X1 , X2 } is an excisive couple. Let A = X1 ∩ X2 .
Then there is an exact sequence
φ∗
ψ∗
. . . −→ Hp (A) −→ Hp (X1 ) ⊕ Hp (X2 ) −→ Hp (X) −→ Hp−1 (A) −→ . . .
called the Mayer-Vietoris sequence of {X1 , X2 } .The homomorphisms are defined by
φ∗ (a) = (i∗ (a), −j∗ (a))
ψ∗ (x1 , x2 ) = k∗ (x1 ) + l∗ (x2 )
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where the maps
i
A −−−→


jy
l
X1


ky
X2 −−−→ X
are inclusions.
Homology of the Sphere
If we let S n ⊂
Rn+1 be the n-sphere (of radius 1 and centre the origin), where n ≥ 0. Then
Z if p = n
n
e
Hp (S ) =
0
if p 6= n
e p (S n ) are called the reduced homology groups of S n . These reduced homology groups are
Here H
e i (X) and Hi (X) are isomorphic.
more readily used and for i > 0 the groups H
e p (S n ) is as claimed for all
To prove the results above we induct on n ≥ 0 to show that H
p ≥ 0.
For n = 0, we have the 0-sphere S 0 , we should note that it consists of two points {1, −1}
and is therefore a discrete two-point space. We can show that the formula holds for n = 0, by
e 0 (X)
the dimension axiom. It should be noted that for a space with n-components, X, then H
is n − 1 copies of .
If we now assume that n > 0 and let a and b be north and south poles of S n . Let X1 =
S n − {a} and X2 = S n − {b}. We have S n = X10 ∪ X20 and X1 and X2 are contractible. We
should also note that X1 ∩ X2 = S n − {a, b} has the same homotopy type as the equator S n−1 .
If we apply the Mayer-Vietoris sequence for reduced homology we obtain the following exact
sequence;
Z
e p (X1 ) ⊕ H
e p (X2 ) −→ H
e p (S n ) −→ H
e p−1 (X1 ∩ X2 ) −→ H
e p−1 (X1 ) ⊕ H
e p−1 (X2 )
H
Since X1 and X2 are contractible, the direct sum terms are both zero, and so
e p (S n ) ∼
e p−1 (X1 ∩ X2 ) ∼
e p−1 (S n−1 )
H
=H
=H
e p−1 (S n−1 ) =
By induction H
and 0 otherwise.
Z if p − 1 = n − 1 and 0 otherwise, therefore Hep(S n) = Z if p = n,
References
S. MacLane, Categories for the Working Mathematician, Springer-Verlag (1971)
S. MacLane, Homology, Springer-Verlag (1991)
J.R. Munkres, Elements of Algebraic Topology, Addison-Wesley (1984)
J.J. Rotman, An Introduction to Algebraic Topology,Springer-Verlag (1988)
V.A. Vassiliev, Introduction to Topology, American Mathematical Society (2001)
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