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Results tagged with at.algebraic-topology
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user 6153
Homotopy theory, homological algebra, algebraic treatments of manifolds.
1
vote
Exact 1- and 2-forms in $R^n$
See http://en.wikipedia.org/wiki/De_Rham_cohomology and in particular De Rham's fundamental theorem. "Closed = exact" for a domain says the De Rham cohomology group is trivial in the appropriate degre …
- 12.6k
6
votes
Spectral sequence
The Koszul complex is defined at http://en.wikipedia.org/wiki/Koszul_complex . In certain cases, one of which is explained there, the Koszul complex is a resolution (typically a free resolution, see h …
- 12.6k
2
votes
tangent sphere bundle over sphere
It's not so simple in general: see the "vector fields on spheres" problem at http://en.wikipedia.org/wiki/Vector_fields_on_spheres . Odd and even dimensions are different in nature because of the Eule …
- 12.6k
36
votes
2
answers
4k
views
Timeline of cohomology (1935 to 1938)
There was a recent question on intuitions about sheaf cohomology, and I answered in part by suggesting the "genetic" approach (how did cohomology in general arise?). For historical material specific t …
- 12.6k
2
votes
Intuition on finite homotopy groups
One approach is to understand this as part of the geometry of universal covering spaces. Such spaces are simply connected (even sometimes contractible), but may have finite groups acting on them suffi …
- 12.6k
1
vote
Time-line until the publicaton of Weil of "Numbers of solutions of equations in finite fields"
The papers by Roquette
http://www.rzuser.uni-heidelberg.de/~ci3/rv.pdf, http://www.rzuser.uni-heidelberg.de/~ci3/rv2.pdf, http://www.rzuser.uni-heidelberg.de/~ci3/rv3.pdf
and
http://www.rzuser.uni- …
- 12.6k
11
votes
Where do all these projection formulas come from?
The first (set theory) formula is generalised in categorical logic to what is called "Frobenius reciprocity" there, and is then part of the handling of the existential quantifier (a natural way to go …
- 12.6k