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Results tagged with homotopy-theory
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user 14447
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
Accepted
Does the reduced Mapping cylinder have the same homotopy type of unreduced Mapping cylinder?
I don't trust anything with degenerate basepoints, but I do think you are right about the homotopy type via the parenthetical "(or only using an explicit homotopy)''. However, I was working in compa …
- 30.4k
3
votes
Does the reduced Mapping cylinder have the same homotopy type of unreduced Mapping cylinder?
I'm lousy at point-set topology, always was, but I don't see that $X$ is closed in $X\wedge I_+$. This is the sort of question a working algebraic topologist does not want to think about. Cofibrant …
- 30.4k
2
votes
Accepted
Path space of a simplicial topological space?
It is sensible to restrict to simplicial {\em based} spaces $X_*$ and then apply the functor $P$ levelwise. This is used, for example, to compare $|\Omega X_*|$ with $\Omega |X_*|$ in The Geometry o …
- 30.4k
7
votes
Accepted
Algebraic $K$-theory of algebras in symmetric spectra: reference
Tom, not precisely sure what you want. I'm guessing you want
to think of a commutative symmetric ring spectrum $R$ and then an
$R$-algebra $A$. Without the extra layer, just using an $S$-algebra $R$ …
- 30.4k
2
votes
On the obstruction of a sequence of simplicial spaces that is levelwise a fibration
Working with spaces meaning spaces, I gave a reasonably strong result along these lines in
Theorem 12.7 of The Geometry of Iterated Loop Spaces http://www.math.uchicago.edu/~may/BOOKS/geom_iter.pdf.
…
- 30.4k
4
votes
Homology of homotopy fixed point spectra
Maybe to expand on the relationship between naive and genuine G-spectra, Tom's answer is stated
in terms of naive G-spectra, but the actual mathematics, the Segal conjecture, is all about genuine G-sp …
- 30.4k
7
votes
Accepted
Does a pointed homotopy equivalence between pointed $G$-spaces which is $G$-equivariant indu...
In the pointed Borel construction, you clearly mean $\wedge$
and not $\times$. Thus $$EG_+\wedge_G X = EG\times_G X/EG\times_G\ast.$$
Out of laziness, I'll assume that your $X$ and $Y$ are of the $G …
- 30.4k
7
votes
Coherent MU_*-Modules
The question is also addressed in Lecture 5 of J.F. Adams ``Lectures on generalized cohomology'' in Springer Lecture Notes in Mathematics Vol 99(1969). Again ancient, but none the worse for that.
Th …
- 30.4k
13
votes
Is a map a homotopy equivalence if its suspension is so?
Since adequate answers to the question asked have been given, I will address the relevant underlying theorems. There are two relevant theorems, both called ``Whitehead's theorem''. One says that a w …
- 30.4k
19
votes
Accepted
Computing homotopies
Sometimes easy geometric pictures have awkward seeming algebraic descriptions.
On pages 6 and 7 of Concise, I gave examples where I both gave a geometric picture
and explicit formulas to make the idea …
- 30.4k
2
votes
The homotopy cofiber of the smash product of two maps of spectra
An axiomatization of exactly how smash products of cofiber sequences
should behave is given in the context of triangulated categories in
my paper
The additivity of traces in triangulated categories …
- 30.4k
9
votes
Accepted
$\mathcal{I}$-functors and infinite loop spaces
You are looking at the telescope of maps $BG_n\longrightarrow BG_{n+1}$ where the coproduct
of the $G_n$ (thought of as categories) has a structure of permutative category. The group
completion prope …
- 30.4k
0
votes
A homotopyish Landweber exact functor theorem
Akhil, short though that paper is, I claim that the proof there is as
complete as it needs to be.
- 30.4k
6
votes
Accepted
units in non-commutative ring spectra
The notation $\Omega^{\infty}$ requires care. Definitions and comparisons for symmetric, orthogonal,
and EKMM spectra are given in Lind's thesis. See https://arxiv.org/abs/0908.1092. For symmetric …
- 30.4k
22
votes
Why study the p-completions of a space?
I'll give an answer from the point of view of an algebraic topologist,
somebody who cares about examples and computations. This all goes way back
and has nothing to do with modern generalities. Fir …
- 30.4k