Questions tagged [homotopy-theory]

Question 1

Consider four paths $f_1$ from $x_1$ to $y_1$, $f_2$ from $x_2$ to $y_2$ , $g_1$ from $x_1$ to $x_2$, and $g_2$ from $y_1$ to $y_2$ in $X$. Assume there is a homotopy $F$ between $f_1$ and $f_2$ and a ...

Question 2

Suppose that $E_0, E_1 \rightarrow M$ are two $k$-dimensional vector bundles over a manifold $M$ classified by maps $\phi_0, \phi_1: M \rightarrow BGL(k)$. If $\phi_0, \phi_1$ are homotopic, then $E_0,...

Question 3

I came up with a vauge question while reading this post. Does the simplicial circle $S^1$ have anything to do with the topology of $\mathbb{C}$-points of some variety? The linked question suggests ...

Question 4

Let $X$ be a topological space and $G \subset \text{Aut}(X)_{\text{top}}$ a finite group acting faithfully on $X$ "nicely enough", where "nicely enough" = that we can form quotient ...

Question 5

Let $F \to E \to B$ be a fibration. In Section 12 of Milnor and Stasheff, they consider a local system over $B$ $$ \{\pi_n(F)\} $$ whose fiber over $b \in B$ is the group $\pi_n(E_b)$. I'm struggling ...

Question 6

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 ...

Question 7

I am trying to do the following exercise: Given a compact, Hausdorff topological space $X$ and a metrizable topological space $Y$ (with metric $d_Y$), let $C(X,Y)$ be the space of continuous functions ...

Question 8

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 ...

Question 9

Following Hatcher's book on algebraic topology (page 410) a Postnikov tower for a path-connected space $X$ is certain commutative diagram together with a sequence on maps $f_n: X \to X_n$ (1) induces ...

Question 10

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 $...

Question 11

This question is a slight extension of (the answer of) this. For some context: So when I learn homotopical algebra (essentially model categories, also Waldhausen categories for K-theory), I get the ...

Question 12

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 $F(s,1)=e_{y_0}$. ...

Question 13

I am trying to understand homotopy fixed points for a finite group $G$, and how they relate to the original space or category. In particular, if we have a functor $X$ from $BG$ (category with one ...

Question 14

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 ...

Question 15

Assume $X$ is a simply connected space and $B\mathbb{S}^1$ the classifying space of the universal bundle $E\mathbb{S}^1\rightarrow B\mathbb{S}^1$. Are any two maps $f_1,f_2: X\rightarrow B\mathbb{S}^1$...

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Questions tagged [homotopy-theory]

Does path concatenation preserve equivalence relation of homotopies?

Vector bundles with equivalent sphere bundles

Simplicial circle in motivic homotopy theory

Correspondence Local Systems & $\Bbb Z \pi_1(A)$-modules and its Compatibility

Do fiberwise homotopy groups define a local system?

Where to find Nardin's lecture notes on stable homotopy theory?

Definition of homotopy (exercise)

Why a certain condition is satisfied in triangulated categories

Postnikov Tower in Hatcher's AT (replacement of maps by fibrations)

Counterexamples (homotopies) [closed]

Homotopies in hammock localization

Well defindness of homotopy map $F(s,t)=h\left(f(s-st)\right)$

Do homotopy fixed points embedd into the original space/category?

Doubt on left Ore conditions

Maps to base space of classifying space

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