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Results tagged with at.algebraic-topology
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user 140013
Homotopy theory, homological algebra, algebraic treatments of manifolds.
7
votes
1
answer
2k
views
Which revolutions in topology and geometry can we expect in the next 20 years? [closed]
In my limited perspective on the history of mathematics, I can name at least two big revolutions in Topology and Geometry (broadly construed): the introduction of Schemes in Algebraic Geometry, and th …
6
votes
1
answer
365
views
Different flavours of Vassiliev Conjecture
There is something that puzzles me about "Vassiliev's Conjecture". I am sure I am missing some detail which is obvious to the community, since there are several tightly related kind of results around …
- 2,152
5
votes
1
answer
346
views
Analogues of Sullivan Theory at a prime for coformality
In rational homotopy theory, one can study the rational homotopy and cohomology categories via an algebraic structure, respectively the rational Lie Algebra model and the Sullivan cdga model.
If I am …
- 2,152
2
votes
0
answers
281
views
Notation for spectral sequences [closed]
Every single spectral sequence I have seen in my life was denoted by $E$. Even when there is more than one spectral sequence, people tend to use the same letter with some workaround (e.g. a fourth sub …
- 2,152
4
votes
0
answers
243
views
Being a product - from homology to topology
The famous Kunneth formula expresses the homology of a product manifold as the tensor product of the two algebras.
Now suppose we know that a manifold $X$ has a decomposition $H_*(X) \simeq A \otimes …
- 2,152
12
votes
Why should an algebraic geometer care about singular / simplicial (co)homology?
The very advantage of algebraic topology over algebraic geometry is the existence of a segment. From a segment it's easy to construct triangles, tetrahedra, and simplices in general. There are at leas …
- 2,152
4
votes
1
answer
182
views
Homotopy coherent space maps induces homotopy coherent chain complex morphisms
It is an elementary fact that a map $f:X \to Y$ between spaces induces a chain complex morphism $f_* : C_*(X) \to C_*(Y) $. This allows one to transfer 1-category theoretic arguments from spaces to as …
- 2,152
4
votes
0
answers
418
views
Persistent homotopy groups
Everybody in algebraic topology loves homology and cohomology, but sometimes we like homotopy groups also, since they detect different things (think about spheres) .
An interesting and recent applicat …
- 2,152
6
votes
2
answers
522
views
Deformation of a diagram preserve the homotopy limit
I have been a bit sloppy in the title, but let me be specific. I stepped again into the subtle difference between homotopy limit and limit in the homotopy category, in the following version.
Suppose y …
- 2,152
1
vote
0
answers
118
views
1-connected infinity groupoids, groupoids and 1-connected spaces
I am exploring a bit the world of groupoids. What I have in mind is that infinity groupoids correspond to spaces. So my first question is the following:
Consider the model category $\infty-Grpd$ of …
- 2,152
1
vote
Definition of Left Operadic Kan Extension for $\infty$-operads
A recommendation: make drawings of cones!
Note that by definition
$$ (X_{/b})_n = Hom_b((\Delta^n)^{\triangleright},X)$$
Where $Hom_b$ means maps that sends the cone point to b. By Yoneda Lemma we c …
- 2,152