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Results tagged with homotopy-theory
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user 10366
Homotopy theory is an important sub-field of algebraic topology. It is mainly concerned with the properties and structures of spaces which are invariant under homotopy. Chief among these are the homotopy groups of spaces, specifically those of spheres. Homotopy theory includes a broad set of ideas and techniques, such as cohomology theories, spectra and stable homotopy theory, model categories, spectral sequences, and classifying spaces.
4
votes
Is it possible to compute coefficients of the formal group of an elliptic curve?
Suppose that $C$ is given by a homogeneous Weierstrass equation $f(x,y,z)=0$. Then there is a unique series $\xi(x)=\sum_{k>0}\xi_kx^k$ such that $\xi(x)=x^3+O(x^4)$ and $f(x,1,\xi(x))=0$. This can …
- 56.9k
12
votes
Ring structure on the $K(1)$-local homotopy of $S^0$ at the prime 2
Here is another method which is in some sense more direct. Consider the diagram
$$ \begin{array}{ccc}
L_{K(1)}S & \xrightarrow{i} & KO \\
i \downarrow & & \downarrow (1,\psi^3) \\
KO & \xrightarr …
- 56.9k
6
votes
Accepted
Homotopy of localisations of colimits
There are two natural finiteness conditions that you might impose on $Y$. The stronger one says that the Morava $E$-theory of $Y$ is finite in each degree, or equivalently that $Y$ is a retract of $L …
- 56.9k
6
votes
Spaces that invert weak homotopy equivalences.
Here is an interesting test case: let $B$ be the Stone-Cech compactification of a set $S$, let $A$ be the underlying set of $B$ with the discrete topology, and let $f$ be the identity map. Then $B$ i …
- 56.9k
8
votes
Accepted
classifying space of linear embeddings
Let $\mathcal{A}$ be your category. This is a skeleton of the category $\mathcal{B}$ of all nonzero finite-dimensional complex Hilbert spaces and isometric embeddings, so $B\mathcal{A}$ is homotopy e …
- 56.9k
6
votes
Accepted
pushout and homotopy
No. Take $A=[0,1]$, $B=S^1$, $C=\text{point}$. There is only one possible choice for $f_1$ and $g_1$. There are many possible choices for $f_0$ and $g_0$, but they are all homotopic. Choose $f_0$ …
- 56.9k
6
votes
Accepted
Pushout of spaces
Take $B_0$ to be an annulus. Let $B_1$ be the space obtained by collapsing the left half of the inner circle to a point, and let $B_2$ be obtained by collapsing the right half of the inner circle to …
- 56.9k
12
votes
Computing the differentials in the Adams spectral sequence
In principle, everything is algorithmically computable, but the proof does not lead to practical algorithms. In practice, you can find some differentials by ad hoc means and then deduce many more dif …
- 56.9k
6
votes
Accepted
Splitting low-dimensional $p$-local CW complexes for large $p$
This result appears in the PhD thesis of Hans-Werner Henn, and in this paper:
@article {MR884630,
AUTHOR = {Henn, Hans-Werner},
TITLE = {Classification of {$p$}-local low-dimensional spectra …
- 56.9k
7
votes
Is higher-order excision related to higher-order cohomology operations?
This is not really an answer but it is an explanation of a point of view that I'd like to advertise. Let $\mathcal{C}$ be a triangulated category. Freyd constructed an abelian category $\mathcal{A}$ …
- 56.9k
6
votes
Accepted
Homotopy type of the geometric realization of a poset
First recall that geometric realisation of posets preserves products: the projections $P\xleftarrow{p}P\times Q\xrightarrow{q}Q$ give a map $(|p|,|q|)\colon |P\times Q|\to|P|\times|Q|$, and it is a st …
- 56.9k
6
votes
Two $E_\infty$ structures on infinite matrices
First note that if $f\colon\mathbb{R}^\infty\to\mathbb{R}^\infty$ is a linear isometric embedding and $u\in O(n)\leq O$ we can define $f_*(u)\in O$ by $f_*(u)(f(x))=f(u(x))$ when $x\in\mathbb{R}^n$ an …
- 56.9k
14
votes
Accepted
Simplest example of non-trivial Toda bracket in spaces
I think that the simplest nontrivial case comes from the maps
$$ S^5 \xrightarrow{\Sigma^2\eta} S^4 \xrightarrow{2\iota} S^4 \xrightarrow{\Sigma\eta} S^3 $$
Both two-stage composites are trivial, so …
- 56.9k
17
votes
Accepted
A homotopyish Landweber exact functor theorem
Here are three methods that I know:
In the case $M_*=(MU_*/I)[S^{-1}]$ (where $I$ is generated by a regular sequence) there is a more direct construction by reducing to the cases $M_*=MU_*/a$ and $M …
- 56.9k
7
votes
Homotopy Extension Property involving mapping cylinder
I'll assume you want the convention where $M_f$ is $(X\times I)\cup Y$ with $(x,0)$ attached to $f(x)$. Now $M_f\times I=(X\times I^2)\cup(Y\times I)$ with $(x,0,t)$ attached to $(f(x),t)$. We want …
- 56.9k