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Results tagged with homotopy-theory
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user 102519
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.
28
votes
Accepted
A refinement of Serre's finiteness theorem on unstable homotopy groups of spheres
$P(\eta) = [i_n,i_n] \circ \eta$, where $[i_n,i_n]: S^{2n-1} \rightarrow S^n$ is the Whitehead product of the identity map with itself. So you are asking if this composite is null.
I don't know if t …
- 11.2k
21
votes
Accepted
Maps out of Eilenberg-Mac Lane Spaces
This question was totally answered by Alex Zabrodsky, right after Haynes Miller proved the Sullivan conjecture. See the paper: "On phantom maps and a theorem of H. Miller", Israel J. Math. 58 (1987), …
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17
votes
Accepted
Solving polynomial equations in spectra?
Here is a simple argument that would show many finite complexes can not be `integral' in your sense.
If $Sq^{2^k}$ acts nontrivially on $H^*(X;\mathbb Z/2)$ then $Sq^{2^{k+1}}$ will act nontrivially o …
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17
votes
Accepted
Is there a "higher Segal conjecture"?
In the 1980's, Chun Nip Lee showed that the Spanier Whitehead dual of (the suspension spectrum of) $K(\mathbb Z/p, n)$ is contractible for $n >1$. (The key case is $n=2$. The idea: view $K(A,n+1)$ as …
- 11.2k
16
votes
Other homotopy invariants?
A hot topic for 20 years starting in the mid 1980's was the exploration of spaces by `probing' them with the spaces $BV$ where $V$ is a group of the form $(\mathbb Z/p)^n$ with $p$ a prime. It is a …
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16
votes
What clues originally hinted at stability phenomena in algebraic topology?
For those whose German is shaky or non-existent, it is fun to copy and paste a couple of paragraphs of Freudenthal's paper into Google translate. The answer to your question emerges. His paper is co …
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15
votes
Accepted
Dyer-Lashof algebra and Steenrod algebra "duality"
The original paper on Koszul algebras, [Stewart Priddy, Koszul resolutions. Trans. Amer. Math. Soc. 152 (1970) 39–60], was, in essence, written to explain this example. Well almost: he was considerin …
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14
votes
Accepted
Applications of equivariant homotopy theory to representation theory
There are decades and decades of algebraic results that use techniques from equivariant homotopy theory. Some examples ...
(1) Quillen's work on ring theoretic aspects of the cohomology of finite gro …
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12
votes
Computation of $\pi_4$ of simple Lie groups
Since you are asking a question about $\pi_4$, lets use a little homotopy theory to think about this, and for starters, we can just ponder compact Lie groups.
Consider first $G=Sp(1)=SU(2)=S^3$. I …
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12
votes
Accepted
Are the AHSS and Adams spectral sequence the same when computing connective Morava K-theory ...
Tyler's comment answers my question. A bit more detail: the Postnikov tower of $k(n)$ is an Adams resolution, because the `bottom class' map $k(n) \rightarrow H\mathbb F_p$ is onto in mod $p$ cohomo …
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12
votes
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Pointed versus unpointed maps into a topological monoid
As is implicitly pointed out in the comments, you really want to assume that $X$ ($=M$) is path connected. And then your analysis is fine. Note that $M$ will then wish to be equivalent to $\Omega BM …
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12
votes
Accepted
Product structure in Milnor exact sequence
Let $P$ be the wedge of all the $X_i$s. Up to homotopy equivalence, $X$ is the homotopy coequalizer of the identity and the shift maps from $P$ to itself. The Milnor exact sequence arises by analyzing …
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11
votes
Why do Chern classes and Stiefel-Whitney classes satisfy the "same" Whitney sum formula?
There are many ways to prove these formulae uniformly, and, of course, you need to start from an appropriate definition of these classes. But as has already been suggested in the comments, pretty muc …
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11
votes
Accepted
Homotopy fibre of composition
This is a fundamental and basic property in homotopy theory, as is the dual statement for homotopy cofibers. These appear as Lemmas 1.2.7 and 1.2.5 in the recently published book More concise algebra …
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11
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Is there an explicit Dold-Thom theorem?
The homology of symmetric products has been studied in lots of ways by many people for 70 years. If one enters `homology of symmetric products' into MathSciNet one gets 480 Math Reviews. A good star …
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