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Results tagged with homotopy-theory
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user 8103
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.
2
votes
Criterion for deloopable based map
Yes. The criterion is that $f:G\to H$ be an $A_\infty$-map with respect to the $A_\infty$-structures on $G$ and $H$. The definition is given in Section 4 of Stasheff's Homotopy associativity of $H$-sp …
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6
votes
Accepted
is $\pi_k(X_1,A_1)$ a direct summand of $\pi_{k}(X_1\vee X_2,A_1\vee A_2)$
The pair $(X_1,A_1)$ is a retract of the pair $(X_1\vee X_2,A_1\vee A_2)$, so the map $\pi_k(X_1,A_1)\to \pi_k(X_1\vee X_2,A_1\vee A_2)$ is a split injection for all $k$. If $k\ge 3$ then the groups a …
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5
votes
Accepted
Homology-Cohomology Pairing
Propositions 17.10 and 17.11 of Switzer's book seem to be examples of things working nicely.
For those who don't have the book to hand: let $E$ be a commutative ring spectrum such that $E_\ast(E)$ is …
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5
votes
Accepted
Integral cohomology operations related to Landweber-Novikov
This is more of a comment than an answer. I just wanted to point out that such an operation, if it were to exist, cannot be stable.
This follows from the answer to Integral cohomology (stable) operat …
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2
votes
Singular complex = cohomology ring + Steenrod operations?
I think the answer is no, due to this answer to this question. There are Moore spaces with the same cohomology rings and module structure over the mod $p$ Steenrod algebra, which nevertheless have dif …
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8
votes
The most general context of Mather's Cube Theorems
You might be interested in the work of Jean-Paul Doeraene: http://math.univ-lille1.fr/~doeraene/
In particular, on pages 8 and 9 of the paper Homotopy pull backs, push outs, and joins, he gives sever …
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2
votes
Existence of homotopy inverses for co-H spaces
The argument you give for the simply-connected case generalizes to nilpotent connected co-H spaces. This is essentially because nilpotent spaces are $H_\ast(-;\mathbb{Z})$-local. See
Hilton, Peter; M …
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1
vote
A question about homotopy dimension
Spheres have this property: If $Y=S^n$ and $X\le Y$ then since the identity homomorphism on $\pi_i(X)$ factors through $\pi_i(S^n)$ for all $i$, we have that $\pi_i(X)=0$ for $i<n$ and $\pi_n(X)=0$ or …
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16
votes
Accepted
Homotopic classification of maps $M \to \mathbb{RP}^n$ where $M$ is a compact orientable $n$...
This seems to have been worked out in the 1960s by Paul Olum, see Section 1 of
Olum, P., Cocycle formulas for homotopy classification; maps into projective and lens spaces, Trans. Am. Math. Soc. 103 …
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2
votes
Classifying maps into homogeneous spaces up to homotopy
Very late, I know, but I just stumbled upon this question and I thought I would point out the reference http://arxiv.org/abs/0808.0024, which gives a nice modern discussion of this problem in the case …
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10
votes
What are the uses of the homotopy groups of spheres?
If you believe that CW-complexes are nice spaces, then two-cell complexes are among the nicest spaces of all. These are spaces of the form $X = S^n \cup_\alpha e^{m+1}$, where $\alpha: S^m\to S^n$ is …
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8
votes
Accepted
Can every element of a homotopy group of a smooth manifold be represented by an immersion?
There exists a simply-connected closed $6$-manifold $M$ with a homotopy class $\alpha\in \pi_4(M)$ which does not contain an immersion. The following argument is due to Diarmuid Crowley, after we real …
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33
votes
What is the 31st homotopy group of the 2-sphere?
One simple observation is that $\pi_{31}(S^2)\cong\pi_{31}(S^3)$, by the long exact sequence of the Hopf fibration.
The homotopy groups $\pi_i(S^3)$ for $i\le 64$ are apparently computed in:
Curtis …
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6
votes
Table of (integral) cohomology groups of K(Z,n)
There are several computations carried out explicitly in the paper
Samuel Eilenberg and Saunders Mac Lane, MR 65162 On the groups $H(\Pi,n)$. II. Methods of computation, Ann. of Math. (2) 60 (1954), …
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27
votes
A possible generalization of the homotopy groups.
Back in the 1940's, Ralph Fox defined something called the torus homotopy group. For a based space $(Y,y_0)$ and natural number $r$, the $r$-dimensional torus homotopy group $\tau_r(Y,y_0)$ is just th …
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