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Results tagged with homotopy-theory
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user 12166
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.
17
votes
Accepted
Does every (co)homology functor (in particular, stable homotopy) factor through chain comple...
No (I mean, not in a triangulated way), otherwise any generalized homology theory of a mod 2 Moore space would be 2-torsion, but this is not true for mod 2 stable homotopy groups (it's well known that …
- 15.2k
3
votes
Motion planning algorithm
I think so, assuming that $X$ is a CW-complex, of course. The evaluation map is a fibration, and $\mathcal M(X)$ is the fiber of the following fibration between mapping spaces
$$\operatorname{map}(X\t …
- 15.2k
5
votes
When are (weak) homotopy equivalence testable on open covers?
Let me offer sufficient conditions in both cases. They follow from the existence of the following two left proper model structures on the category of topological spaces, and the well-known gluing lemm …
- 15.2k
1
vote
Accepted
Components of a loop space, semidirect products, and multiplicativity
I think the answer is ' yes' for trivial reasons. If the loop space were a topological group the map $\varphi$ would be a homeomorphisms and the group structure on the target could be transferred to t …
- 15.2k
8
votes
Accepted
Is the derived category of abelian groups a subcategory of the stable homotopy category?
I've found the following somewhat intricate way of answering Q1 in the affirmative. Any complex in $D(Ab)$ quasi-isomorphic to a graded abelian group. Hence, it is enough to consider complexes concent …
- 15.2k
4
votes
Singular complex = cohomology ring + Steenrod operations?
It depends of what structure you want to consider on the complex of singular cochains. If you want to look at it just as a complex, then the cohomology groups are enough. If you want it as a different …
- 15.2k
9
votes
Accepted
homotopy transfer for sheaves of algebras
If $R$ is a commutative $k$-algebra, a quasi-coherent sheaf of dg-$k$-algebras on $\operatorname{Spec}R$ would be just a dg-$R$-algebra $A$. It's known that there need not be any $R$-linear A-infinity …
- 15.2k
3
votes
Examples of Brown (co)fibration categories that are not Quillen model categories?
Michael Weiss considers in his paper "Hammock localization in Waldhausen categories" the Waldhausen category of $G$-CW-spectra ($G$ a discrete group) where cofibrations are $G$-CW-inclusions and weak …
3
votes
Examples of Brown (co)fibration categories that are not Quillen model categories?
If you don't want to be forced to enlarge you may be interested in tha category of topological spaces and proper maps. They form a cofibration category (in the sense of any of the available definition …
12
votes
Accepted
Signs in the unstable homotopy groups of spheres
OK, after having made so many stupid comments, I felt obligated to remember what I knew about unstable homotopy theory in order to try to say something meaningful.
Recall that $\pi_7(S^4\vee S^4)\con …
- 15.2k
9
votes
Accepted
What is the homotopy type of a free simplicial ring?
Let $R[-]$ be the free $R$-module functor, from sets to $R$-modules, and $T_R$ the free (tensor) $R$-algebra functor, from $R$-modules to $R$-algebras. The free $R$ algebra functor from sets to $R$-al …
- 15.2k
23
votes
Accepted
Homology of the fiber
As usual, there's no loss of generality in assuming that $f$ is the inclusion of a subspace $X\subset Y$, replacing $Y$ with the homotopy equivalent mapping cylinder of $f$ if necessary. By your assum …
- 15.2k
5
votes
Which homotopy 2-types are H-spaces?
I just wanted to add to Tyler Lawson's answer that all the maps $\beta\colon K(G,1)\rightarrow K(A,3)$ ($G$ and $A$ abelian and no action of $G$ on $A$) satisfying his additivity condition are loop ma …
- 15.2k
7
votes
Accepted
Does the right adjoint of a Quillen equivalence preserve homotopy colimits?
The homotopy colimit functor $Ho(D^I)\rightarrow Ho(D)$ is the left adjoint of the constant diagram functor $Ho(D)\rightarrow Ho(D^I)$. Quillen equivalences induce Quillen equivalences between diagram …
- 15.2k
9
votes
Accepted
Choice of fibrations is like a choice of a basis of a module
I guess that 'the letter' is meant to be Grothendieck's Pursuing Stacks, which started as a letter to Quillen (as one can read in the document) and then evolved in a kind of book/diary addressed to th …
- 15.2k