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Results tagged with homotopy-theory
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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.
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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31
votes
Accepted
An equivariant map from sphere to a Lie group of lower dimension which is not null homotopic?
The Blakers-Massey element in $\pi_6(S^3)\cong\mathbb{Z}_{12}$ can be represented by such a map. This is done explicitly on page 3 of the paper https://arxiv.org/abs/math/0501091, published as
Abres …
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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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24
votes
Accepted
H-space structures on non-sphere suspensions?
If $Y$ is a connected CW-complex of finite type which is both an H-space and a co-H-space, then $Y$ has the homotopy type of $S^1$, $S^3$, $S^7$ or a point. This is a result of Robert West:
Robert W. …
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23
votes
What is classified by generalised Eilenberg MacLane spaces?
To answer your first question, take a look at the reference
Gitler, Samuel, Cohomology operations with local coefficients, Am. J. Math. 85, 156-188 (1963). ZBL0131.38006.
In particular, Theorem 7.18 …
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19
votes
Unstable homotopy groups of spheres beyond Toda's range
By following Ryan's leads I've been able to find references computing $\pi_{n+k}(S^n)$ for $20\leq k \leq 30$ at the prime $2$ (and in some cases at odd primes as well). I thought I'd post these as an …
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17
votes
Homotopy pullback of a homotopy pushout is a homotopy pushout
This is Mather's second cube theorem, see Theorem 25 in
Mather, Michael, Pull-backs in homotopy theory, Can. J. Math. 28, 225-263 (1976). ZBL0351.55005.
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16
votes
What are Homotopy rings good for?
You may be interested in reading about $\Pi$-algebras, which are graded abelian groups possessing all the primary algebraic structure of $\pi_*(X)$, for $X$ a pointed and connected topological space. …
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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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11
votes
Accepted
Alternate proofs of Quillen's theorem on formal group laws and MU
Quillen's proof of this in the paper
Elementary proofs of some results of cobordism theory using Steenrod operations. Advances in Math. 7 1971 29–56 (1971)
does not make use of Adams spectral sequen …
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10
votes
Are loop spaces of homotopically equivalent spaces homotopically equivalent?
The answer is yes always, provided $f\colon\thinspace (X,x_0)\to (Y,y_0)$ is a pointed homotopy equivalence of pointed spaces (meaning that the homotopies $g\circ f \simeq 1_X$ and $f\circ g\simeq 1_Y …
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10
votes
Accepted
Why does $\iota_4^2 \in H^8(K(\mathbb Z/2,4);\mathbb Z/2)$ not come from $H^8(K(\mathbb Z/2,...
It is true, and follows from results of Browder on the mod 2 Bockstein spectral sequence for $K(\mathbb{Z}/2,4)$. (We can replace $4$ by any even integer $k$ and conclude that $\iota_k^2$ ia not the r …
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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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10
votes
Coincidences amongst classifying spaces and Eilenberg Mac-Lane spaces
There is a principal bundle
$$Sp(1)\to S^{4n+3} \to \mathbb{H}P^n$$
for each $n$, which on passing to the limit shows that $$\pi_i(\mathbb{H}P^\infty)\cong\pi_{i-1}(Sp(1))=\pi_{i-1}(S^3)$$
for each $i …
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10
votes
Accepted
smooth homotopy on exotic R^4
Yes, since smooth maps which are (continuously) homotopic are always smoothly homotopic. See Kosinski's "Differential Manifolds", Theorem III.2.5 and Corollary III.2.6.
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