Definition:Cohomology Group
Jump to navigation
Jump to search
Definition
Let $X = \struct {S, \tau}$ be a topological space.
Let $n \in \Z_{\ge 0}$ be a non-negative integer.
Let $f: X \to Y$ denote a continuous mapping from $X$ to another topological space $Y$.
The cohomology groups $\map {H^n} X$ of $X$ are variants of the homology groups of $X$, but with the characteristic property that, given $f: X \to Y$, the corresponding homeomorphisms $f^*$ run from $\map {H_n} Y$ to $\map {H^n} X$ rather than the other way round.
 | This article is complete as far as it goes, but it could do with expansion. In particular: Whoosh. If anyone know their way around this area, a proper formal definition of this is in order. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by adding this information. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Expand}} from the code.If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page. |
Motivation
Cohomology groups arise naturally in the statement of the Poincaré Duality Theorem for Manifolds.
They are also important because the cohomology groups of a topological space $X$ can also be given ring structure.
This makes them more powerful in algebraic topology than homology.
Examples
Also see
- Results about cohomology groups can be found here.
Sources
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): cohomology
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): cohomology