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Results tagged with homotopy-theory
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user 2625
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.
39
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3
answers
6k
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Why do finite homotopy groups imply finite homology groups?
Why does a space with finite homotopy groups [for every n] have finite homology groups? How can I proof this [not only for connected spaces with trivial fundamental group]? The converse is false. $\ma …
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19
votes
3
answers
5k
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What determines a model structure?
It is easy to prove that a model structure is determined by the following classes of maps (determined = two model structures with the mentioned classes in common are equal).
cofibrations and weak eq …
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17
votes
2
answers
3k
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Why does homotopy behave well with respect to fibrations and homology with respect to cofibr...
(I apologize that this is a vague question).
I seems to me somehow that homotopy groups behave well with respect to (Serre)-fibrations. For example you get a long exact sequence of homotopy groups fr …
- 2,782
11
votes
1
answer
936
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Analogue to Serre spectral sequence for cofiber sequences and homotopy
(This is a follow-up question to this one).
As it is nicely outlined in an answer to this question, homotopy groups behave well with respect to (Serre)-fibrations and (co)homology groups behave well …
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10
votes
4
answers
1k
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Singular complex = cohomology ring + Steenrod operations?
Fix a prime $p$ and consider everything mod $p$. Steenrod operations arise somehow from the loss of information passing from the singular complex of a space to its cohomology ring. Are they exactly th …
- 2,782
6
votes
3
answers
1k
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Why are spectra indexed over the natural numbers?
A spectrum is a sequence $X_0,X_1,...$ of spaces together with structure morphisms $\Sigma X_n\to X_{n+1}$. To get the usual model for the stable homotopy category based on the category of spectra, on …
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3
votes
1
answer
467
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Closure of the homotopy relation for a simplicial set
Define a reflexive relation on the set of zero-simplices of a simplicial set $A$ by saying that $x\sim y$ iff there is a one-simplex $h$ with $0$-face $y$ and $1$-face $x$. This is not an equivalence …
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2
votes
3
answers
2k
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Distributivity / commutativity of pushouts and pullbacks
What does it exactly mean to say that in a certain category pushouts and pullbacks "commute"? Is it the same to say that they "distribute"?
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1
vote
1
answer
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Is there a map of spectra implementing the inverse of the Thom isomorphism?
In the top answer to the question "Is there a map of spectra implementing the Thom isomorphism?" it is explained (with a reference to Rudyaks book) that from a rank $r$ vector bundle $\mu:V\to X$, a s …
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