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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.
6
votes
What does $\mathrm{Conf}_n(M)^{h S_n}$ look like?
[UPDATE: Connor Malin's answer is definitely better than mine, but I will leave mine here in case the approach turns out to be useful for some other purpose.]
Put $X=\operatorname{Conf}_n(M)$ and $Y=\ …
- 56.9k
6
votes
Accepted
Euler class in center of mod 2 Morava K-theory?
Yes, there are various ways to see this. One approach is to say that in $K(n)$-theory we always have $ab-ba=v_nQ(a)Q(b)$ for a certain Bockstein-type operation $Q$ of odd degree $2^n-1$, so any eleme …
- 56.9k
7
votes
Homotopy of Brown-Gitler spectra
This is not an answer but perhaps gives interesting context. The Brown-Gitler spectra are (up to suspension) wedge summands in the spectrum $\Sigma^\infty_+\Omega^2S^3$. Up to homotopy there is a un …
- 56.9k
9
votes
Accepted
Are there “nice” explicit representations of infinite order elements in $\pi_{4n-1}(S^{2n})$
You can write an explicit formula as follows. Consider a pair of finite-dimensional inner product spaces $U$ and $V$. We can define a map
$$ f\colon S(U\oplus V) \to S(U\oplus\mathbb{R})\vee S(V\opl …
- 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
3
votes
The complex $K$-theory of the Thom spectrum $MU$
For a more self-contained answer, let $L$ be the tautological line bundle over $\mathbb{C}P^\infty$. This gives a class $x=[L]-[1]\in\widetilde{K}^0(\mathbb{C}P^\infty)$. It is standard that $K^0(\m …
- 56.9k
7
votes
Accepted
Homotopy groups of $K(n)$-local $E_n$-modules are $L$-complete
The identity map of $S/I$ satisfies $I_n^r.1_{S/I}=0$ for $r\gg 0$, so $I_n^r.\pi_k(M\wedge S/I)=0$ for $r\gg 0$, so $\pi_k(M\wedge S/I)$ is $L$-complete. The spectra $S/I$ can be assembled into a to …
- 56.9k
10
votes
Accepted
Can we prove that any number of algebraic data cannot be a complete invariant of a homotopy ...
I'll answer the corresponding question for the homotopy category $\mathcal{S}$ of spectra. I doubt that this makes much difference, but I have not checked the details. We can choose a list $X_0,X_1, …
- 56.9k
2
votes
Accepted
Geometric realization of a poset
Put $P_0=\{1,\dotsc,n-k-1\}$ and $P_1=\{n-k,\dotsc,j-1\}$ and $P_2=\{j,\dotsc,n\}$, so $[n]=P_0\amalg P_1\amalg P_2$ with each $P_i$ nonempty. Any subset $U$ can be decomposed as $\coprod_{i=0}^2U_i$ …
- 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
13
votes
Why the stable module category?
One reason is just that $\text{stab}_{kG}(k,M)_*=\widehat{H}^{-*}(g;M)$ (the Tate cohomology of $G$ with coefficients in $M$). I think that Tate invented Tate cohomology for applications in Class Fie …
- 56.9k
7
votes
Accepted
If $A_\ast$ has a Künneth theorem, then is $A$ a module over Morava $K$-theory?
If you want $K(h)_*$ to be strong monoidal, then you need the target category to be the category of graded $K(h)_*$-modules, not the category of graded abelian groups. Thus, I assume you are really a …
- 56.9k
11
votes
Are Chern classes well defined up to contractible choice?
I think that the best approach is to use the construction in the 1993 paper "Algebraic cycles and infinite loop spaces", which I learned about from this answer by Jeremy Hahn. To explain the context, …
- 56.9k
10
votes
Accepted
How to construct $X \oplus \Sigma X$ from $X \oplus \Sigma X \oplus \Sigma X \oplus \Sigma^2...
The cofibre of $0\oplus 1\oplus 1\oplus 1$ on $X=Z\oplus \Sigma Z\oplus\Sigma Z\oplus\Sigma^2Z$ is $Z\oplus\Sigma Z=Y$.
- 56.9k
15
votes
Cobordism cohomology of Lie groups
Firstly, for any space $X$ we have an Atiyah-Hirzebruch spectral sequence
$$ H^p(X;MU^q) \Longrightarrow MU^{p+q}(X). $$
The differentials are always torsion-valued, essentially because the higher sta …
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