Search Results
| Search type | Search syntax |
|---|---|
| Tags | [tag] |
| Exact | "words here" |
| Author |
user:1234 user:me (yours) |
| Score |
score:3 (3+) score:0 (none) |
| Answers |
answers:3 (3+) answers:0 (none) isaccepted:yes hasaccepted:no inquestion:1234 |
| Views | views:250 |
| Code | code:"if (foo != bar)" |
| Sections |
title:apples body:"apples oranges" |
| URL | url:"*.example.com" |
| Saves | in:saves |
| Status |
closed:yes duplicate:no migrated:no wiki:no |
| Types |
is:question is:answer |
| Exclude |
-[tag] -apples |
| For more details on advanced search visit our help page | |
Results tagged with homotopy-theory
Search options answers only
not deleted
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.
6
votes
What can be said about the map $K(n)_\ast(X) \to K(n)_\ast(\tau_{\leq n}X)$ when $X$ is a fi...
Let me make more concrete a comment I already wrote. In [Adv. Math. 201 (2006), 318-378], my example 2.22, illustrating a theorem just before it when $n=1$, says that, for any spectrum $Y$, there is a …
- 11.2k
5
votes
To what extent is homological localization determined by its values on $K(G,n)$'s?
You ask two different questions. Regarding the first, there are easy counterexamples: e.g. $\widetilde K(1)_*(S^3) \neq 0$, but $\widetilde K(1)_*(K(\pi_n(S^3),n)) = 0$ for all $n$.
- 11.2k
4
votes
Two models for the classifying space of a subgroup via the geometric bar construction
One way to think about this is as follows. One has a topological groupoid $\mathcal G$ with objects $G/H$ and morphisms $G \times G/H$. Similarly one has the groupoids $\mathcal G^{\prime}$, and $\m …
- 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 …
- 11.2k
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
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), …
- 11.2k
9
votes
Accepted
Koszul duals of n-fold loop spaces
I am not sure if this gives what you want, but maybe it is:
I went in the other direction in a paper The McCord model for the tensor product of a space and a commutative ring spectrum, in Progress in …
- 11.2k
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 …
- 11.2k
9
votes
Accepted
Complex cobordism and Chern numbers
It is a key result that the composite
$$ MU_* \xrightarrow{h} H_*(MU;\mathbb Z) \xrightarrow[\sim]{\Phi^{\vee}}H_*(BU;\mathbb Z),$$
where $\Phi^{\vee}$ is the dual of the Thom isomorphism $\Phi$, agr …
- 11.2k
3
votes
Snaith splitting for operads in spectra?
The best proofs of the `Snaith splitting' that you write down at the beginning, uses the $J^{\mathcal O}$ construction on spectra with units. Here is a complete proof:
Apply $J^{\mathcal O}$ to the …
- 11.2k
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 …
- 11.2k
5
votes
Identify the sphere bundle of a complex line bundle $BD_{2n}\to BU(1)$
As is described in any standard group cohomology textbook, if $Q$ is a group and $A$ is an abelian group, then elements of $H^2(Q;A)$ correspond to central group extensions $A \rightarrow G \rightarro …
- 11.2k
5
votes
Compatibility of Definitions of Universal G-bundles
There is a subtlety here that I think the classical literature doesn't deal with well. Consider the following three statements for a principle $G$--bundle $E \rightarrow B$:
(a) $E$ is a terminal obj …
- 11.2k
3
votes
What is nullification with respect to an Eilenberg-MacLane space?
This question is more fun at the space level: see [Neisendorfer, Joseph A.
Localization and connected covers of finite complexes. The Čech centennial (Boston, MA, 1993), 385–390, Contemp. Math., 181, …
- 11.2k
4
votes
Homology of a loop-suspension space and action of $\mathcal{D}_1$-operad
Your space $D_1(X)$ is equivalent to Ioan James' space $JX$ investigated in the 1950's. It is quite easy to directly show that the homology of this is the tensor algebra on the reduced homology of $X …
- 11.2k