Description

The Cohomology is the dual theory to Homology. Dual in the sense that we define Chain complex as earlier but now we apply the covariant functor . This has the effect of flipping the arrows around. Note that this is not equivalent to the categorical Dual (Category theory), where the arrows of a commutative diagram are flipped. Because the -functor is different, the Cohomology is not just a dual homology but has distinct properties. (like a “good product”)

Cohomology is a contravariant functor from chain complexes to groups. This can be seen from the construction below, where we started with a chain complex. A map between chain complexes (e.h. the boundary map) induces a map on cohomology going the other way around. A similar thing holds for a map between two topological spaces. The cohomology likes to go in the opposite direction of the arrow. To explain:

Motivation

We look at the following problem. Say, for a space we have a map assigning a line to each point. Can we realise by a vector field ? Yes! By the Lifting criterion of a Covering space this is possible exactly if for every possible curve the induced curve on turns around an even amount of time. So, the process of determining the answer is to pushforward every possible curve on and check if the number is even.
We can do better! We can define a cohomological object which measures whether a curve turns around the circle an even or odd times. Covariance of the Cohomology now means that the object likes to move opposite to an arrow, giving us an object in whose vanishing tells us exactly, when the problem has a solution! Without any curves!

  • (1.1) A classifying map seems to be a map which assigns structure to a space . How is it defined? A classifying map is an elementary construction. I should go on Wikipedia but now this is not really required.

We define the Cohomology algebraically, meaning without any relationship to topology, for now.

Definition (Differential like in Hatcher)

Let be an Abelian group. Let be a Chain complex with boundary map/differential . We define the dual cochain group as the group of Homeos to a group . This gives us a dual chain complex with a codifferential . Cohomology is defined as

The differential is never explicitly stated. Is it between and or between and ? The only way to check is to visit the books of Hatcher and Bredon.

  • What exactly is a dual and why is it a dual only if . (Sebastian said so). The “dual” issue is that there are two kinds of dual at play. The hom functor and the category theory dual
  • Why is the sign needed? Is there a dual reason for homology. The sign is needed to make the construction categorical but is not so easy to legitimise otherwise
  • What is the point of a dual construction, when the construction is not exactly dual? See above
  • Should the script be an error-free object, i.e. do you wish to be notified if we find errors? Yes!

Other sources like to use a sign in the differential. This is the natural construction one would obtain if we we wanted to stay consistent with category theory. It also apparently makes computations easier. Hatcher didn’t choose this route because category theory has no place in his book and he didn’t want to make the differential overly complex.

Definition (Different differential)

Like above we define the chain cocomplexes as . Differently from above, we now use the codifferential We define the homology as previously

Properties

Relationship to Homology

For any chain complex of free Abelian groups and every Abelian group , there is a natural surjective map and the Short exact sequence is split.

The map is important, i.e. the proof should be read. Short proof: An element is represented by a . Using we can show that this map descends to a homomorphism .
To show splitting, we study the s.e.s. . Since all groups here are free Abelian there is a right inverse to and the sequence splits. This split allows us to define a projection from to . We show surjectivity by defining a right inverse of called . This is obtained by chaining some maps together and then showing that is the identity.

Revelations:

  • If you see a at the end, try inserting , then do a quotient.
  • We always have . To show that something is in the kernel show that it is in the image instead by using .
  • Exact split sequences seem to be extremely useful in calculation