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Results tagged with at.algebraic-topology
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user 184
Homotopy theory, homological algebra, algebraic treatments of manifolds.
66
votes
1
answer
2k
views
Is there an octonionic analog of the K3 surface, with implications for stable homotopy group...
The infamous K3 surface has many constructions in many fields ranging from algebraic geometry to algebraic topology. Its many properties are well known. For this question I am really interested in the …
45
votes
Difference between represented and singular cohomology?
This is a good question because it really hits on a subtle issue. It turns out that Johannes and Ben are both correct and incorrect at the same time unless we settle some very subtle issues. Let me ex …
35
votes
1
answer
1k
views
Finding the octonionic analog of the K3 surface, via (almost) hyperkahler geometry?
The K3 manifold is an amazing object in mathematics which plays an important role in several fields ranging from the study of smooth 4-manifolds to algebraic geometry to differential geometry and beyo …
35
votes
Accepted
Maps inducing zero on homotopy groups but are not null-homotopic
Consider ordinary singular cohomology with varying coefficients. You can look at the short exact sequence of abelian groups:
$$0 \to \mathbb{Z}/2 \to \mathbb{Z}/4 \to \mathbb{Z}/2 \to 0$$
This give …
34
votes
2
answers
5k
views
Example Wanted: When Does Čech Cohomology Fail to be the same as Derived Functor Cohomology?
I want to know exactly how derived functor cohomology and Cech cohomology can fail to be the same.
I started worrying about this from Dinakar Muthiah's answer to an MO question, and Brian Conrad's com …
31
votes
3
answers
2k
views
Is the counit of geometric realization a Serre fibration?
Recall that a Serre fibration between topological spaces is a map which has the homotopy lifting property (HLP) for all CW complexes (equivalently for all disks $D^k$). The Serre fibrations are the fi …
30
votes
Accepted
"Homotopy-first" courses in algebraic topology
I was a heavily involved TA for such a graduate course in 2006 at UC Berkeley.
We started with a little bit of point-set topology introducing the category of compactly generated spaces. Then we move …
29
votes
4
answers
1k
views
Which stable homotopy groups are represented by parallelizable manifolds?
The Pontryagin-Thom construction allows one to identify the stable homotopy groups of spheres with bordism classes of stably normally framed manifolds. A stable framing of the stable normal bundle ind …
28
votes
Accepted
Nilpotence of the stable Hopf map via framed cobordism
Answer Summary
Let $\eta$ be the framed 1-manifold which is the Lie group framing on the circle and let $\nu$ be the Lie group framing on $S^3 = Spin(3)$. I am probably going to conflate these framed …
26
votes
1
answer
1k
views
Why are quasitopological spaces needed in sheaf theoretic approaches to the h-principle?
Recently I have been learning more about the h-principle and in particular the methods of "continuous sheaves". In many treatments of this I see people using "quasi-topological spaces" and I am trying …
26
votes
2
answers
2k
views
Loop Spaces as Generalized Smooth spaces or as Infinite dimensional Manifolds?
There are two ways to define smooth mapping spaces and I want to know how they compare.
Let's take the concrete special case of free loops spaces. I think this is the most studied example so will pro …
25
votes
4
answers
3k
views
A Peculiar Model Structure on Simplicial Sets?
I'm wondering if there is a Quillen model structure on the category of simplicial sets which generalizes the usual model structure, but where every simplicial set is fibrant? I want to use this to do …
24
votes
6
answers
2k
views
Simplicial model of Hopf map?
The Hopf fibration is a famous map $S^3\to S^2$ with fiber $S^1$, which is the generator in $\pi_3(S^2)$. We can model this map in terms simplicial sets by taking the singular simplicial sets of these …
22
votes
4
answers
2k
views
Functorial Whitehead Tower?
The Whitehead tower of a (pointed) space is a tower of spaces which successively kills the bottom homotopy groups. The first two spaces can be constructed functorially (at least for suitably nice spac …
20
votes
Accepted
Every Manifold Cobordant to a Simply Connected Manifold
Assume that $M^n$ has $\pi_1$ finitely generated (Edit: and n>3). Choose a generator. We will construct (using surgery) a cobordism to $M'$ which kills that generator, and by induction we can kill all …