homology

homology

Homology

is the general name for a number of

functors

from

topological spaces

to

abelian groups

(or more generally modules over a fixed ring). It turns out that in most reasonable cases a large number of these (

singular homology

,

cellular homology

, Morse homology,

simplicial homology

) all coincide. There are other generalized homology , but I won’t consider those. There are also related

cohomology

theories which serve the same purpose with slightly different machinery.

In an intuitive sense, homology measures “holes” in topological spaces. The idea is that we want to measure the

topology

of a space by looking
at sets which have no

boundary

, but are not the boundary of something else. These are things
that have wrapped around “holes” in our topological space, allowing us to detect those “holes.”
Here I don’t mean boundary in the formal topological sense, but in an intuitive sense. Thus a loop
has no boundary as I mean here, even though it does in the general topological definition. You
will see the formal definition below.

Perhaps the simplest form of homology to visualize, and to work with in practice, is simplicial homology. It is based on computing the

homology groups

of a

simplicial complex

(generally a finite one). However, it is generally nontrivial to show that a space of interest is

homeomorphic

to a simplicial complex, and it can also be difficult to apply more advanced methods such as

spectral sequences

when working with simplicial homology. Singular homology is

similar

: it is in some sense a

continuous

version of simplicial homology, and it does not suffer from these problems.

Singular homology is defined as follows: We define the standard

n

-

simplex

to be the subset

Δ

n

=

{

(

x

1

,

…

,

x

n

)

∈

ℝ

n

|

x

i

≥

0

,

∑

i

=

1

n

x

i

≤

1

}

of

ℝ

n

. The

0

-simplex is a point, the

1

-simplex a

line segment

, the 2-simplex, a

triangle

,
and the 3-simplex, a

tetrahedron

.

A

singular

n

-simplex

in a topological space

X

is a continuous map

f

:

Δ

n

→

X

.
A

singular

n

-chain

is a formal linear combination (with integer coefficients) of a
finite number of singular

n

-simplices. The

n

-chains in

X

form a group under formal

addition

,
denoted

C

n

⁢

(

X

,

ℤ

)

.

Next, we define a

boundary operator

∂

n

:

C

n

⁢

(

X

,

ℤ

)

→

C

n

-

1

⁢

(

X

,

ℤ

)

. Intuitively, this is just taking
all the faces of the simplex, and considering their images as simplices of one lower

dimension

with the appropriate sign to keep

orientations

correct. Formally, we let

v

0

,

v

1

,

…

,

v

n

be the vertices of

Δ

n

, pick an order on the vertices of the

n

-

1

simplex, and let

[

v

0

,

…

,

v

^

i

,

…

,

v

n

]

be the face spanned by all vertices other than

v

i

, identified
with the

n

-

1

-simplex by

mapping

the vertices

v

0

,

…

,

v

n

except for

v

i

, in that order,
to the vertices of the

(

n

-

1

)

-simplex in the order you have chosen.
Then if

φ

:

Δ

n

→

X

is an

n

-simplex,

φ

⁢

(

[

v

0

,

…

,

v

^

i

,

…

,

v

n

]

)

is the map

φ

, restricted to the face

[

v

0

,

…

,

v

^

i

,

…

,

v

n

]

, made into a singular

(

n

-

1

)

-simplex by the identification with the standard

(

n

-

1

)

-simplex I defined above.
Then

∂

n

⁡

(

φ

)

=

∑

i

=

0

n

(

-

1

)

i

⁢

φ

⁢

(

[

v

0

,

…

,

v

^

i

,

…

,

v

n

]

)

.

It is a simple exercise in reindexing to check that

∂

n

∘

∂

n

+

1

=

0

.

For example, if

φ

is a singular

1

-simplex (that is a path), then

∂

⁡

(

φ

)

=

φ

⁢

(

1

)

-

φ

⁢

(

0

)

.
That is, it is the

difference

of the

endpoints

(thought of as 0-simplices).

Now, we are finally in a position to define homology groups. Let

H

n

⁢

(

X

,

ℤ

)

, the

n

homology
group of

X

be the

quotient

H

n

⁢

(

X

,

ℤ

)

=

ker

⁢

∂

n

im

⁢

∂

n

+

1

.

The association

X

↦

H

n

⁢

(

X

,

ℤ

)

is a functor from topological spaces to abelian groups, and the maps

f

*

:

H

n

⁢

(

X

,

ℤ

)

→

H

n

⁢

(

Y

,

ℤ

)

induced by a map

f

:

X

→

Y

are simply those induced by

composition

of an singular

n

-simplex with the map

f

.

From this definition, it is not at all clear that homology is at all

computable

. But, in fact, homology is often much more easily computed than

homotopy groups

or most other

topological invariants

. Important tools in the calculation of homology are

long exact sequences

, the

Mayer-Vietoris sequence

, cellular homology, spectral sequences, and

homotopy invariance

.

Some examples of homology groups:

H

m

⁢

(

ℝ

n

,

ℤ

)

=

{

ℤ

m

=

0

0

m

>

0

.

This

reflects

the fact that

ℝ

n

has “no holes”

Consider the space

ℝ

⁢

P

n

, real projective space, which is

ℝ

n

+

1

∖

{

0

}

modulo the

relation

that

(

x

0

,

…

,

x

n

)

≡

λ

⁢

(

x

0

,

…

,

x

n

)

for every nonzero

λ

. For

n

even,

H

m

⁢

(

ℝ

⁢

P

n

,

ℤ

)

=

{

ℤ

m

=

0

ℤ

2

m

≡

1

(

mod

2

)

or

n

>

m

>

0

0

m

≡

0

(

mod

2

)

,

n

>

m

>

0

⁢

or

⁢

m

≥

n

,

and for

n

odd,

H

m

⁢

(

ℝ

⁢

P

n

,

ℤ

)

=

{

ℤ

m

=

0

⁢

or

⁢

n

ℤ

2

m

≡

1

(

mod

2

)

or

n

>

m

>

0

0

m

≡

0

(

mod

2

)

,

n

>

m

>

0

⁢

or

⁢

m

>

n

.

Title

homology

Canonical name

Homology

Date of creation

2013-03-22 13:14:41

Last modified on

2013-03-22 13:14:41

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

17

Author

mathcam (2727)

Entry type

Definition

Classification

msc 55N10

Synonym

singular homology

Related topic

SimplicialComplex

Related topic

GeometryOfTheSphere

Related topic

BettiNumber

Related topic

HomologyChainComplex

Related topic

CohomologyGroupTheorem

Defines

singular n-chain

Defines

singular n-simplex

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