# History of cohomology theory

I saw this post. And I already posted it on Math stack exchange, but since someone recommended this site, I'm refining it and posting it again. And I understand that the mathematical object called cohomology came from Poincaré duality. But I would like to know the details of the history of cohomology itself. So, if you know, I would appreciate it if you could leave an answer. Or, I would appreciate it if you could recommend a book related to the history of cohomology with as few formulas as possible.

• Are you interested specifically in cohomology or homology would also be of interest? Mar 5 at 11:15
• @MikhailKatz I'm interested in cohomology and homology Mar 6 at 7:34

A History of Algebraic and Differential Topology, 1900 - 1960 by Jean Dieudonné describes, among other things, the development of (co)homology in great detail.

Cohomology theories were introduced independently by A. Kolmogorov, Ueber die Dualitat im Aufbau der kombinatorischen Topologie, Mat. Sbornik, vol. 1, 1936, 97-102,

and by J. W. Alexander, On the chains of a complex and their duals, Proc. Nat. Acad. Sci, 21, 1935, 509-512, complete exposition in A.. of Math., 37, 1936, 698-708.

Remark. One of the comments mentions Poincare duality. It is only modern formulation of duality that uses cohomology. Poincare himself stated it in terms of homology.

• Of course, one might mention Gauss-Green-Stokes as precursors. Feb 8 at 21:45
• @Paul Garrett: Yes, and especially Maxwell's Treatise on electricity and magnetism. But formal definition is due to Alexander and Kolmogorov. Feb 9 at 14:01
• The OP has clearly indicated that he is interested in both homology theory and cohomology theory. Attributing this to Kolmogorov and Alexander when Poincare introduced the theory over 30 years earlier is bizarre to say the least. Mar 26 at 14:07
• Mikhail Katz: Alexander and Kolmogorov were the first to formally introduce COhomology theory. Mar 26 at 14:35

A key tool in developing homology or cohomology theories is that of a chain complex. Dieudonne writes in his History of Algebraic and Differential Topology:

"The first occurrence of a chain complex is to be found in Poincare's papers of 1900" (page 4).

Dieudonne also mentions some antecedents in the 19th century:

"After an unsuccessful attempt in 1895 to give a genuine mathematical formulation to the intuitive ideas of Riemann and Betti, probably inspired by earlier work of the nineteenth century on polyhedra and the Euler formula, [Poincare] restricted himself to compact triangulated spaces ... From this beginning the evolution of homolog went through series of steps, the description of which constitutes Part 1 of this book."

Dieudonne's introduction is probably a good place to start if one wants to understand the history of (co)homology theories.