**Yoshifumi Tsuchimoto**

Let us denote by the set of all differentiable maps from to . A so-called ``de Rham cohomology'' of is computed as a cohomology of a complex

We see that:

Actually, the dimension of the 0 -th cohomology is related to a number of the connected component of . The dimension of the -st cohomology is related to a number of the `hole' of . Cohomology is then a good tool to obtain numbers (``invariants'') of geometric objects.

Cohomology also arises as ``obstructions''. Indeed, the de Rham cohomology of the tells us a hint about ``which functions are integrable'', etc.

In this talk we give a definition and explain some basic properties of cohomologies. But before that, we first deal with some category theory.

- A collection
of
**objects**of . - For each pair of objects
, a set
**morphisms**. - For each triple of objects
,
a map(``composition (rule)'')

- unless .
- (Existence of an identity) For any
,
there exists an element
such that
- (Associativity)
For any objects
, and for any morphisms
,
we have

- If and , then .
- If , then .
- If , then the power set .
- If is a family of elements of indexed by an element , then .

- If , then .
- If is a subset of , then .
- If , then the ordered pair is in .
- If , then and are in .
- If is a family of elements of indexed by an element , then we have .

In this text we always assume the following.

For any set , there always exists a universe such that .

support of $f$ is compact

Then:

- Compute the cohomology group of the following complex.
- Compute the cohomology group of the following complex.