Questions tagged [homotopy-theory]
Two functions are homotopic, if one of them can by continuously deformed to another. This gives rise to an equivalence relation. A group called a homotopy group can be obtained from the equivalence classes. The simplest homotopy group is the fundamental group. Homotopy groups are important invariants in algebraic topology.
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Correspondence Local Systems & $\Bbb Z \pi_1(A)$-modules and its Compatibility
Let $X$ be a topological space and $G \subset \text{Aut}(X)_{\text{top}}$ a finite group acting faithfully on $X$ "nicely enough", where by "nicely enough" I mean such that we can ...
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Where to find Nardin's lecture notes on stable homotopy theory?
There are lectures on YouTube "Introduction to stable homotopy theory" by Denis Nardin . Apparently, these are recordings of a course given at University of Regensburg in 2021. The lecturer ...
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Why a certain condition is satisfied in triangulated categories
Let $\mathcal T$ be a triangulated category and let $\cal S$ be a triangulated subcategory.
Say that a morphism $A\xrightarrow \alpha B$ is in $\Sigma\subset \mathrm{Mor}(\cal T)$ if, in the ...
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Counterexamples (homotopies) [closed]
I know that if $X,X',Y,Y'$ are topological spaces such that $X,X'$ are homotopically equivalent, $Y$ and $Y'$ are homotopically equivalent, then $X\times Y$ and $X'\times Y'$ and also $X\sqcup Y$ and $...
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Well defindness of homotopy map $F(s,t)=h\left(f(s-st)\right)$
Let $I=[0,1]$, $h:(X,x_0) \to (Y,y_0)$ and $[f] \in \pi_1(X_0)$. Can I construct the following map $$F(s,t)=h\left(f(s-st)\right):I \times I \to Y?$$
Here, $F(s,0)=h \circ f(s)$ and $F(s,1)=e_{y_0}$. ...
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Doubt on left Ore conditions
Let $\cal C$ be a category and let $\Sigma$ be a class of morphisms in $\mathrm{Mor}\cal (C)$. Those are the left Ore conditions given in my course:
given morphisms $f:x\to y$ and $g:y\to z$, if at ...
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Constructing an explicit nullhomotopy for $N:T^2\to S^2$
Let $(\theta,\varphi)\in[0,2\pi)\times[0,2\pi)$ be angular coordinates on the torus of revolution with the standard identifications $(\theta,\varphi)\sim(\theta+2\pi,\varphi)\sim(\theta,\varphi+2\pi)$....
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Gradient bound for $C^1$ function in unit disk [duplicate]
For a function $f : D^n \to \mathbb{R}$, $f \in C^1(D^n) \bigcap C(\overline{D^n})$, suppose $||f||_{L^\infty} \le 1$, we want to show $\inf | Df | \le c$, for all such $f$.
A known result https://www....
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Indicating Discrepance between $\text{Ho}(\text{Fun}(I, \text{Top}))$ and $\text{Fun}(I, \text{Ho(Top)})$ by comparing their Homotopies
Let $I$ some finite diagram category, eg something very simple, say $\bullet \rightarrow \bullet\rightarrow \bullet$ or $\bullet \leftarrow \bullet\rightarrow \bullet$ or a square of $\bullet$'s or ...
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Relation between Homotopy extension property and exact sequences
I am starting to study homotopy theory and there's a relation between the Homotopy extension property (HEP) and exact sequences that I'm not understanding.
For context let me first give the ...
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Curiosity on definition of triangulated category
In the lectures, the teacher stated that a triangulated structure on an additive category $\mathcal T$ consists of: $\require{AMScd}$
an invertible functor $[1]:\mathcal T\to \mathcal T$;
a ...
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Continuity of the homotopy between [f*f']and f(0) shown by Hatcher for the fundamental groupoid [closed]
In the book Algebraic Topology by Allen Hatcher, he defines the fundamental group but seems to have omitted the proof that the homotopy defined between $f_{t}*\bar{f_{t}}$ and the constant path at $f(...
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Weak homotopy equivalence without a true homotopy equivalence (Whitehead’s theorem counterexample)
I'm currently trying to understand some of the subtleties of Whitehead’s theorem.
I'm looking for an example of two spaces $X$ and $Y$ such that:
$X$ is a CW complex,
$Y$ is not a CW complex, and
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Why is the fiber product in the definition of a Segal spaces a homotopy fiber product?
I was reading the paper "A model for the homotopy theory of homotopy theory" by Charles Rezk, and in the definition of a Segal space (Def. 4.1, page 11) he first defines $W\colon\Delta^\...
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A supplement to "FROM FREGE TO GÖDEL A Source Book in Mathematical Logic, 1879-1931" for developments on type theory and computation
This is a cross-posting from Computer Science SE.
(Some background - Its actually the second time I am posting this question here. At the first time I was advised to post it on 'Histoy of maths SE', ...
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homotopy colimit in $Spc_{Quillen}$ and $Spc_{Strom}$?
I am reading the construction of classifing space for a topological group G, which should classify the prinicipal bundles.
If we consider the $(\infty,1)$-category of spaces $S$, then $G$ as a ...
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Compatibility of Homotopy with (Co)limits in $\infty$-categories
I'm reading Ravenel's survey What is $\infty$-categiry... and have problems to grasp the essence the stament from Chapter 8 on colimits of $\infty$-category of spaces:
Pleasant feature of $\infty$-...
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Left adjoint to precomposition with inclusion is not a Quillen left adjoint for injective model structure
I am learning about model categories, and came up with the following (wrong) proof.
Lets assume the injective model structure exist on $\mathrm{Fun}(I,C),\mathrm{Fun}(J,C)$ for some suitable model ...
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Topology of the d-dimensional cobordism category
I am currently reading a paper by Galatius-Madsen-Tillman-Weiss named The homotopy type of the cobordism category. In section 2, the authors define the $d$-dimensional cobordism category $\mathcal{C}...
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Failure for Existence of Homotopy Equivalence between Colimit and Homotopy Colimit [closed]
Let $C$ be an ordinary category containing all small colimits with choosen class $W \subset \text{Mor}(C)$ of weak equivalences. For concrete running example take eg $C=\text{Top}$ (or some nice ...
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Separation of the space via homotopy theory
The Jordan Curve Theorem states that any topological circle (spaces homeomorphic to $S^1$) in the euclidian plane, or in the $2$-sphere, separates it. A proof given by Munkres in his book uses only ...
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Does the linking number change under diffeomorphism?
Let $M^m,N^n\subset\mathbb{R}^k$ be smooth, compact, oriented and disjoint surfaces of $\mathbb{R}^k$ such that $m + n = k - 1$. Then, the linking number $\text{link}(M,N)$ of $M$ and $N$ is the ...
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Finite homotopy groups of THH(Z)
In the paper Topological Hochschild Homology and Integral p-adic
Hodge Theory by Bhatt-Morrow-Scholze, the lemma 2.5 claims that $\pi_i(THH(\mathbb{Z}))$ is finite when $i>0$. I know this result is ...
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Realization of path homotopy
I am reading "complex analysis" by Theodore W. Gamelin, and was stuck by the following statement
The idea of the proof is that any continuous deformation of $\gamma_{0}$ can be
realized as ...
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Cohomology of $B(U/O)$
For computing the cohomology groups of another space, I need $H_3(B(U/O))$ and $H_4(B(U/O))$ with $\mathbb{Z}$ or $\mathbb{Z}_2$ coefficients. Note that by Bott periodicity, $B(U/O) \simeq Sp/U$. I ...
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