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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.
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
10
votes
Accepted
Delooping maps between H-spaces
No, it is not: $S^3$ admits uncountably many loop space structures (c.f. Rector "Loop structures on the homotopy type of $S^3$"), but only $12 (= \vert \pi_6(S^3) \vert)$ H-space structures (c.f. Jame …
- 18.2k
5
votes
free homotopy groups -- when do they exist?
If you are thinking about π1, the action is just that by conjugation in this group, so is trivial iff the fundamental group is abelian.
I think the easist place to see the nontriviality of this actio …
- 18.2k
7
votes
classifying space of linear embeddings
I believe it is (weakly) contractible, for the following amusing reason. Let me write $\mathcal{C}$ for the category described in the question. Direct sum gives a functor $\mu : \mathcal{C} \times \ma …
- 18.2k
1
vote
Classifying spaces of topological categories
Let $L_0$ and $L_1$ be two linear orders over $X$ (I think of these as being the etale space). Let $L'_0 = L_0 \times [0,1)$, which is an etale space over $X \times [0,1]$; let $L'_1 = L_1 \times (0,1 …
- 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
5
votes
1
answer
762
views
Bousfield-Kan spectral sequence with local coefficients
Let $F : \mathcal{D} \to \mathbf{Top}$ be a diagram of topological spaces. A local system of coefficients $M$ on $\mathrm{colim}_\mathcal{D} F$ pulls back to a local system $M_d$ on $F(d)$ for each $d …
- 18.2k
17
votes
2
answers
1k
views
Signs in the unstable homotopy groups of spheres
Let $\mathbb{HP}^2$ denote the quaternionic projective plane. According to
A note on $\mathcal{E}(\mathbb{HP}^n)$ for $n\leq 4$, N. Iwase, K-I. Maruyama, S. Oka, Math. J. Okayama Univ. 33 (1991) , 16 …
- 18.2k
22
votes
Accepted
Group completion theorem
The statement that $M \to \Omega BM$ is a weak equivalence when $M$ is a group-like topological monoid is indeed easier: the map $EM = B(M \wr M) \to BM$ is then a quasi-fibration, has geometric fibre …
- 18.2k
12
votes
Quasifibrations and homotopy pullbacks
It rather depends what you mean by quasi-fibration. The most useful resource I know for questions about quasi-fibrations is this message of Goodwillie, posted to the APGTOP mailing list in 2001.
- 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
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
2
votes
Cobordism and finite sheeted covers of manifolds
Its not true in complex cobordism. In Quillen's paper ``Elementary proofs of some results of cobordism theory using Steenrod operations", Section 4, he computes the complex cobordism class of a princi …
- 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