Cohomology Reversing arrows

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.

Reversing the Arrows

In the above diagram the boundary map goes from lines to points for example. each line maps to exactly 2 points.

 

When we reverse the direction of the boundary map then each point maps to multiple lines.

cochain circle

So we transpose the matrix rather than invert it.

 


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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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