Cohomology

One way to introduce cohomology is to start with homology and reverse the arrows. Can we begin with a chain complex consisting of a sequence of matrices and invert these matrices to get a cochain complex?

There are lots of problems with this:

As an example lets use the chain complex of a flat disk:

The determinant of this matrix is zero so we can't invert it.

chain circle

So the inverse arrows are not just free groups which can be represented by matrices as with homology.

More about reversing the arrows here.

Dual Space

The dual vector space is the set of all s V -> R where

      [ x1 ]
V* =[ a1 a2 a3 ]   V = | x2 |
      [ x3 ]
where a linear functional maps a vector into its underlying real space. This can be shown as a row (covector) times a column (vector):
    [ x1 ]
[ a1 a2 a3 ] * | x2 | = R
    [ x3 ]
If we choose the real value we are mapping to is zero then a vector in a space will map to a plane in its cospace. inner product

Using representable functors

We do this by replacing the free groups by their mapping into set (the integers). This gives a contravariant mapping, see 'representable functor'.

Here we add mappings into the integers Z.

Mappings such as these are contravariant

cochain

For instance take disc as example.

In order to create a representable functor we create a map which assigns an integer value to each point. This is a map from each point to a scalar value, we can represent this as a co-vector.

From this we can derive a map which assigns an integer value to each line and so on, going in the reverse direction.

cochain circle
So, for example, the values at each point might represent voltages or heights or somthing like that.

a=6
b=2
c= -5

The lines would then represent voltage differences, or height differences etc. ab=4
ac=11
bc=7

Calculating Cohomology

So we introduce functions from the elements of the homology (points, lines,triangles ...) to some other quantities. cohomology chain

 

Concepts


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see also:
Correspondence about this page

Book Shop - Further reading.

Where I can, I have put links to Amazon for books that are relevant to the subject, click on the appropriate country flag to get more details of the book or to buy it from them.

flag flag flag flag flag flag Mathematics++: Selected Topics Beyond the Basic Courses (Student Mathematical Library) Kantor, Ida.

Chapters:

  1. Measure
  2. High Dimensional Geometry
  3. Fourier Analysis
  4. Representations of Finite Groups
  5. Polynomials
  6. Topology

Chapter 6 - Topology. Contains a relatively gentle introduction to homology.

 

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