Relations With Classifying Spaces and Group Cohomology
If X is a connected cell complex with homotopy groups πn(X) =0 for all n ≥ 2, then the universal covering space T of X is contractible, as follows from applying the Whitehead theorem to T. In this case X is a classifying space or K(G,1) for G = π1(X).
Moreover, for every n ≥ 0 the group of cellular n-chains Cn(T) (that is, a free abelian group with basis given by n-cells in T) also has a natural ZG-module structure. Here for an n-cell σ in T and for g in G the cell g σ is exactly the translate of σ by a covering transformation of T corresponding to g. Moreover, Cn(T) is a free ZG-module with free ZG-basis given by representatives of G-orbits of n-cells in T. In this case the standard topological chain complex
where ε is the augmentation map, is a free ZG-resolution of Z (where Z is equipped with the trivial ZG-module structure, g m = m for every g ∈ G and every m ∈ Z). This resolution can be used to compute group cohomology of G with arbitrary coefficients.
Read more about this topic: Covering Space
Famous quotes containing the words relations with, relations, spaces and/or group:
“As death, when we come to consider it closely, is the true goal of our existence, I have formed during the last few years such close relations with this best and truest friend of mankind, that his image is not only no longer terrifying to me, but is indeed very soothing and consoling! And I thank my God for graciously granting me the opportunity ... of learning that death is the key which unlocks the door to our true happiness.”
—Wolfgang Amadeus Mozart (1756–1791)
“I have no wealthy or popular relations to recommend me.”
—Abraham Lincoln (1809–1865)
“Surely, we are provided with senses as well fitted to penetrate the spaces of the real, the substantial, the eternal, as these outward are to penetrate the material universe. Veias, Menu, Zoroaster, Socrates, Christ, Shakespeare, Swedenborg,—these are some of our astronomers.”
—Henry David Thoreau (1817–1862)
“Jury—A group of twelve men who, having lied to the judge about their hearing, health, and business engagements, have failed to fool him.”
—H.L. (Henry Lewis)