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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.
30
votes
Non-examples of model structures, that fail for subtle/surprising reasons?
I like the following example because it is very close to the origins of homotopy theory (and also because I worked on it at the beginning of my career): proper homotopy theory. Objects are topological …
- 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
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
15
votes
Accepted
Cohomology theories for spaces vs cohomology theories for spectra
The category of cohomology theories on pointed CW-complexes is not equivalent to the stable homotopy category. The latter projects onto the former, and this projection induces a bijection on isomorphi …
- 15.2k
12
votes
Accepted
How to detect if a simplicial set is the nerve of a groupoid?
The answer is sort of well known. A simplicial set $X$ is the nerve of a groupoid if and only if any $n$-horn has a unique filler for all $n\geq 2$. The horn $\Lambda^k[n]$ is the simplicial subset of …
- 15.2k
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
11
votes
Accepted
Homotopy type of tensors of Moore spectra
For $p\neq 2$ the answer is yes. It's an easy computation usinghomology decompositions in the sense of Eckmann–Hilton the well known exact sequence
$$\operatorname{Ext}(A,\pi_{n+1}(X))\hookrightarrow …
- 15.2k
11
votes
Accepted
The second stable homotopy group
I love this question! I've enjoyed thinking of it. Below, I show why the sequence splits always.
Let $X\to Y=K(H_1(X,\mathbb{Z}/2),1)$ be the map inducing the identity in $H_1(-,\mathbb{Z}/2)$. By nat …
- 15.2k
10
votes
Accepted
Fubini theorem for hocolim
This property holds actually for right derivable categories in the sense of:
MR2729017 Reviewed Cisinski, Denis-Charles Catégories dérivables. (French) [Derivable categories] Bull. Soc. Math. France 1 …
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10
votes
Connectivity of suspension-loop adjunction
Indeed, as John Klein shows, the map is $(2k+2)$-connected. Let me offer an alternative proof of the fact that, for $X$ a $k$-connective spectrum, $k\geq 0$, the homomorphism $\pi_i\Sigma^\infty\Omega …
- 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 …
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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 …
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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
9
votes
do spectra have diagonal maps?
The smash product is not a categorical product, so you can't speak of diagonal map, in the same way as you don't have a natural diagonal map $M\rightarrow M\otimes M$, for $M$ an abelian group or vect …
- 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 …
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