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Results tagged with at.algebraic-topology
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user 8824
Homotopy theory, homological algebra, algebraic treatments of manifolds.
4
votes
Cohomology of the classifying space of $Ss(4m)$
I don't think the cohomology is known. As an indication of this I would refer to the 2007 paper "On the Stiefel–Whitney classes of the representations associated with Spin(15)" by Mamoru Mimura and Te …
- 12.9k
7
votes
Accepted
Image of J in the classical Adams Spectral Sequence
The image of $J$ is pretty easy to see in the Adams $E_2$ term: it consists of the elements along the vanishing line, plus, in dimensions 8k-1, of the towers that end near the vanishing line.
This ide …
- 2,821
5
votes
Realizing $\mathcal{A}(2)//\mathcal{A}(1)$ by a finite spectrum
As pointed out by John Rognes, this answer is not correct (it misses the $Sq^4$ from the bottom class to the class in dimension $4$). Sorry for the confusion.
=========== previous answer ============ …
- 1,554
11
votes
Accepted
Adams-Novikov spectral sequence at p = 2
I don't think anybody knows how to compute this $E_2$-term efficiently (not just at the prime $2$). I would love to be proved wrong on this, of course.
So far the only documented, algorithmic method …
- 1,554
11
votes
Intuition behind Thom class
The idea behind the Thom isomorphism
$\beta:H^iX \rightarrow H^{n+i}(DE,SE)$
is implicit in the formula
$$\int_{\sigma_{n+i}} \beta(\alpha_i) = \int_{X\cap \sigma_{n+i}} \alpha_i$$
Here $\sigma_{n+i} …
- 1,554
12
votes
Geometric interpretation of families in the stable homotopy groups of spheres
This has been a burning question for quite some time, but not much is known. Surely, people believe that the next layer (i.e. the $\beta$-family) should also admit a geometric description, although as …
- 1,554
15
votes
What are the best known results for the stable homotopy groups of spheres?
Computing $\pi_\ast(S)$ is a tedious business that to this day can only be done "by hand", i.e. by humans. The $p=2$ computation up to dimension 64 was completed by Kochman (see his SLNM book) with la …
- 1,554
4
votes
Accepted
Computing squaring operations in the Adams spectral sequence
I think for $k>0$ currently no-one knows an efficient algorithmic way to compute the $Sq^k$, e.g., from a minimal resolution. (The $Sq^0$ is easy since it is induced by the "Frobenius" map on $A_\ast$ …
- 1,554
3
votes
"The Z/2-cohomology functor from Top to GrVecSpaces factors through Unstable-A-Mod, and this...
My impression is that Haynes Miller uses the term "algebraic category" in the informal sense in his notes (but I may be wrong). So let me answer this informal version first: one could argue that there …
- 1,554
7
votes
Accepted
Which sets of Stiefel-Whitney characteristic numbers can be realized as coming from a manifold?
The Steenrod operations on mod 2 cohomology imply the vanishing of some characteristic numbers. Specifically, if $p(w_1,w_2,\ldots)\in H^k(M^n;Z/2)$ for $k\lt n$ then $0=\langle \sum_{i+j=n-k}u_i{\rm …
- 1,554
5
votes
Accepted
Morava's "Motives and cell bundles"?
Google knows about this preprint ("A theory of base motives") which seems related:
http://folk.uio.no/rognes/yff/morava.pdf
- 1,554