This post isn’t going to be a complete recount of sheaf cohomology. There is a succinct article fulfilling that purpose provided by the Stacks Project… 2,036 more words

## Tags » Cohomology

#### Intuition of Homology, Cohomology and Homotopy

Homology 同调 is better than Euler Characteristic to differentiate manifolds with holes (eg. Torus, Klein Bottle, …)

Cohomology: 上同调

#### Filling the Details #4: Maps of Eilenberg MacLane Spectra induces stable cohomology operations

First of all, sorry for not having written lately. I’ve just done the GRE and GRE math examinations, together with the TOEFL. I really hope the result will be good, since I’m going to apply in the USA! 780 more words

#### Filling the Details #3: Homotopy Fibrations and Long Exact sequence in Cohomology

Currently I’m reading the paper of Prof. Peter Teichner On the Signature of four-manifolds with universal covering spin, and I was stuck on the following passage: 539 more words

#### The Gelfand-Fuchs cocycle

Let be a vector field on the Euclidean line . Expressed in a coordinate the vector field is . The logarithm of the component value at each point of the vector field is the fundamental Euclidean invariant… 365 more words

#### Random Exercise #2: Problem 11-C in Milnor's Characteristic Classes

Todays’ random exercise will be the pretty famous exercise 11-C one can find in Milnor’s Book. Let me recall the statement:

Let and be compact oriented manifolds with smooth embedding . 393 more words

#### Random Exercise #1 "Isomorphism in cohomology induces Isomorphism in Homology"

I’ve decided to start collecting here some random exercises arose during discussions with my colleagues (usually during the weekly pizza-time) and I found interesting.

Let be a continuous map between topological spaces, and some coefficients. 293 more words