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Results tagged with homotopy-theory
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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.
2
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1
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316
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Is it true, the space of embeddings segments is homotopy equivalent to the subspace of all l...
Consider the space of smooth embeddings of the segment in the plane with the compact-open topology. Denote by X the quotient space obtained from the equivalence relation $a \sim b$ if and only if $\ma …
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7
votes
0
answers
305
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Can every finitely presented group be realized as a fundamental group of a compact four-dime...
Every finitely presented group is realized as a fundamental group of a two-dimensional complex (a simple exercise on Van Kampen's theorem). I was told that a two-dimensional complex can be well embedd …
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1
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1
answer
203
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Is the decomposition of the homotopy type of a complex into a product and into a smash produ...
Is it true that if $A_1\times A_2\times ... \times A_n = B_1\times B_2\times .. \times B_m$, where $A_i, B_j$ are homotopy types of connected complexes not decomposable into a product, then the multi …
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16
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2
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698
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Is the decomposition of the homotopy type of a complex into a bouquet unique?
Is it true that if $A_1 \vee A_2 \vee .. \vee A_n = B_1 \vee B_2 \vee .. \vee B_m$, where $A_i, B_j$ are homotopy types of complexes not decomposable into a bouquet, then the multisets $A_i$ and $B …
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5
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1
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446
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Can we prove that any number of algebraic data cannot be a complete invariant of a homotopy ...
The statement in the title seems to be generally accepted as true, but I have not seen proof. They are?
The strict formulation I have in mind is the following. By an algebraic category we mean the cat …
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8
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2
answers
1k
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Does the isomorphic of the fundamental groups imply the existence of a mapping inducing an i...
A pair of continuous mappings $f \colon X \to Y$ and $g \colon Y \to X$ is called $\pi_1$-equivalence if they induce mutually inverse isomorphisms of fundamental groups. Spaces are called $\pi_1$-equi …
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4
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0
answers
103
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Dugger's theorem for enriched model categories
We know that a combinatorial model category has a small presentation.
Is an enriched version of this theorem known? The closest I could find is: Guillou, May - Enriched model categories and presheaf c …
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10
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1
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663
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Do elements of every order occur in homotopy groups of spheres?
It is known from Serre's classical result that every p-torsion occurs in the homotopy groups of every sphere. Is it known: do elements of every order occur in homotopy groups of spheres?
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26
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1
answer
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Spheres with the same homotopy groups
What is known about the existence of other pairs of spheres (such as $S^2$ and $S^3$) whose homotopy groups coincide starting from some index.
A sufficient condition for this is the existence of a fib …
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3
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1
answer
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How to get by with only functorial cylindrical objects?
In the excellent "A Handbook of Model Categories" (2021), cofibrant and fibrant homotopies are defined exactly as it seemed natural to me: immediately through functorial cylindrical objects / path spa …
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15
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1
answer
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If homotopy groups of spaces are identical, then stable ones are also identical?
Is it true that if pointed spaces $X, Y$ have the same homotopy groups $\pi_n(X) \cong \pi_n(Y)$, then they have the same stable homotopy groups $\pi^S_n(X) \cong \pi ^S_n(Y)$?
In particular, is this …
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6
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230
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"Whenever we have some interesting invariant of spaces, we try to cook up a space that repre...
In his essay Classifying Spaces Made Easy
Baez writes:
We've seen this trick a couple of times lately, and it's actually a
big theme in homotopy theory: whenever we have some interesting
invariant of …
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3
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2
answers
496
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Can the loops in the definition of the fundamental group be considered injective?
Let $\mathrm{С}$ be some class of topological spaces that includes at least all subspaces of $\mathbb{R}^n $. Further we are in the category $\mathrm{С}_{*}$ (the category of point spaces; all continu …
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3
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0
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212
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CW-complexes that cannot be homotopically compressed
Definition. A CW-complex $A$ can be elementary compressed to a CW-complex $B$ if there is a deformation retraction $A \to B$ or a quotient map by a contractible subcopmlex $A \to B$ (meaning according …
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1
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0
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132
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What are the obstacles for a complex to be a space of loops?
It is known that any space of loops is an H-space. So my question has two parts:
What are the obstacles for a complex to be an H-space? Is there any hope to somehow reasonably classify/characterize a …
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