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Results tagged with homotopy-theory
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user 318
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.
8
votes
Accepted
Is Morava K-theory of a classifying space of a compact Lie group a Noetherian ring?
As in the paper Local Gorenstein duality in chromatic group cohomology by Pol and Williamson that Drew Heard pointed out, this can be proved by mimicking the proof of the Quillen--Venkov theorem. Howe …
- 18.2k
3
votes
Accepted
Group completion of a monoid (braid groups)
This is explained in my paper "Group-Completion", local coefficient systems, and perfection, Q. J. Math. (Quillen Memorial Issue) 64 (3) (2013) 795-803. A sufficient condition is for the monoid to be …
- 18.2k
3
votes
On the homological dimension of a Borel construction
I think $f: M /\!\!/ G \to B\Gamma$ cannot be nontrivial on $\mathbb{Q}$-(co)homology in degrees beyond the dimension of $M$, because I think one can find a factorisation
$$f_* : H_*(M /\!\!/ G ; \mat …
- 18.2k
9
votes
Are finite $G$-spectra idempotent complete?
I think the correct setting to look at this question is that of
W. Lück, "Transformation groups and algebraic K-theory". Lecture Notes in Mathematics, 1408. Mathematica Gottingensis. Springer-Verlag, …
- 18.2k
10
votes
Accepted
Homological stability and Waldhausen A-theory
I don't think that you can deduce homological stability of the coinvariants from the Serre spectral sequence as you suggest. But this precise situation was studied in my paper "An upper bound for the …
- 18.2k
3
votes
Accepted
Density of compactly-supported homeomorphisms
I think this is true. It suffices to prove the
Lemma. Given an orientation-preserving homeomorphism $h$ of $\mathbb{R}^n$ there is a compactly-supported homeomorphsim $h_1$ which agrees with $h$ on th …
- 18.2k
10
votes
0
answers
291
views
A certain semi-simplicial space
I would like to understand whether the following construction has been studied. Let me model the pointed sphere as $S^n = I^n / \partial I^n$, and let $\Omega^n (-) := map((I^n, \partial I^n), (-, *)) …
- 18.2k
11
votes
Accepted
Homotopy groups of Diffeomorphisms of punctured d-dim ball
Let me assume that you are interested in diffeomorphisms which also fix the boundary of the ball. If $F_n(D^d)$ denotes the space of $n$ distinct ordered points in $D^d$, then there is a fibration seq …
- 18.2k
5
votes
Transfer map of simplicial sets
The statement is not true (in topological spaces or simplicial sets). The composition $f'' \cdot f'$ will only be multiplication by $d$ "up to higher Atiyah--Hirzebruch filtration".
For an explicit …
- 18.2k
19
votes
Accepted
Product-like structures on spheres
Your condition determines the map $a_1 \vee a_2 : S^n \vee S^n \to S^n$ on the $n$-skeleton, so the question is when does this extend over the $2n$-cell of $S^n \times S^n$. The $2n$-cell is by defini …
- 18.2k
9
votes
Accepted
Counterexamples for strengthening Whitehead's theorem?
It seems to me that the stronger statement is true. If $f$ is only an isomorphism on homotopy in degrees $* \leq n$ then the homotopy fibre $F$ is $(n-1)$-connected and its Hurewicz map is an isomorph …
- 18.2k
16
votes
Accepted
Topological Grothendieck Construction
This is a standard irritation. The issue is that $Top$ is not a category internal to $Top$, because it doesn't have a space of objects (and I don't mean for set-theoretic reasons), so what do you mean …
- 18.2k
5
votes
Is there an analog of compactified moduli spaces(/stacks) for smooth manifolds?
I have a piece of juvenilia on this topic, considering the case of zero-dimensional submanifolds. See Section 9 of
O. Randal-Williams, Embedded cobordism categories and spaces of submanifolds,
IMRN …
- 18.2k
16
votes
Accepted
Are all unstable homotopy groups of $U(n)$ torsion?
Being an $H$-space, $U(n)$ has the rational homotopy type of a product of odd-dimensional spheres. As we know its cohomology, these are $S^1 \times S^3 \times S^5 \times \cdots \times S^{2n-1}$. In pa …
- 18.2k
26
votes
Accepted
Homology theory represented by Madsen-Tillmann spectra
This is an exercise in understanding the Pontrjagin--Thom correspondence. The group $\pi_k(MTO(n) \wedge X_+)$ is in bijection with tuples of
a $(n+k)$-manifold $M$,
an $n$-dimensional vector bund …
- 18.2k