# Tagged Questions

Two functions are homotopic, if one of them can by continuously deformed to another. This gives rise to an equivalence relation. A group called homotopy group can be obtained from the equivalence classes. The simplest homotopy group is fundamental group. Homotopy groups are important invariants in ...

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### Induced homomorphisms on fundamental group

Define the map $f : S^{1} \times S^{1} \to S^{1}$ with $f(x,y) = xy$ and $g : S^{1} \times S^{1} \to S^{1} \times S^{1}$ with $g(x,y) = (xy,x)$ where $x, y \in \mathbb{C}$ on the unit circle $S^{1}$. ...
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### $\Omega$ of a homotopy cofiber sequence

What is an example of a homotopy cofiber sequence $$X\to Y\to Z$$ of well-pointed connected CW-complexes such that the associated sequence of loop spaces $$\Omega X\to \Omega Y\to\Omega Z$$ is not ...
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### Zeroth homotopy group of the space $O(3)/H$

Short version of the question: Is $\pi_0(O(3)/C2) = Z_2$ as $\pi_0(O(3)) = Z_2$? Here $C_2$ is a cyclic group of order 2. Long version or the question: The zeroth homotopy group describes the ...
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### Intuition for chain homotopy via tensor products

An approach to chain homotopies, alternative to the usual boundary relation, uses the monoidal (closed) structure of $\mathsf{Ch}_\bullet(R\mathsf{Mod})$ with $R$ a commutative ring. In particular, a ...
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### Is the stable homotopy group of sphere a commutative ring? If not, are there easy examples?

Is the stable homotopy group of spheres a commutative ring? If not, are there easy examples? In the Adams spectral sequence converging to the stable homotopy group of spheres, it seems that any page ...
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### Homotopy equivalence?

Can someone explaine what this means mathematicaly : "Let us denote by $h: X\rightarrow Y$ a homotopic equivalence map for which $h|_{Y}$ is the identity " Remark: $Y$ is include in $X$ Please ...
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### use homology groups to obtain minimal cell structure

On Hatcher's book Algebraic Topology, Section 4.C Prop. 4C.1, for a simply-connected CW-complex $X$, if $H_*(X;\mathbb{Z})$ is known as a graded module over $\mathbb{Z}$, then the minimal cell ...
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### what does Homotopy Tell?

What is the Homotopy geometrically?And what is path-homotopy? If two real valued functions are homotopic to each other, then what can we say about the geometry behind this? It need not be equal ...
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### $S^1$ a p-local complex?

Let $p$ be a prime. Is $S^1$ a p-local CW-complex? Meaning, for any reduced homology theory $\overline{E}_*$, do we have $\overline{E}_*(S^1)=\overline{E}_*(S^1) \otimes_{\mathbb{Z}} \mathbb{Z}_{(p)}$...
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### conjugation of Lie groups and homotopy group

Let $G$ be a Lie group. Let $\phi, \psi \in \pi_n(G)$. Consider $\theta \in \pi_n(G)$ defined by $\theta(x):= \phi(x)\psi(x)\phi(x)^{-1} \in G$, where we use multiplication and inversion in $G$ in ...
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### Extending a Homotopy from $X \times [0,1]$ to $X \times \mathbb{R}$

In Jeffery M. Lee's Manifolds and Differential Geometry Exercise 1.77: For smooth manifolds $X$ and $Y$, show that if $f_0: X \rightarrow Y$ and $f_1: X \rightarrow Y$ are $C^r$ homotopic then ...
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### The Seifert-Van Kampen theorem as a push-out

My question concerns the proof of the Seifert-Van Kampen theorem. The version of such a statement that interests me is the following. Let $X$ be a topological space, and $U, V\subseteq X$ two path-...
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### Adjoining an $(n+1)$-cell is an $n$-equivalence

Suppose $X$ is a topological space and $x_0 \in X$. Let $$X' = X \cup e^{n+1}$$ be obtained from adding a $(n+1)$-cell (so $X'$ is the pushout of the map $\partial e^{n+1} \to e^{n+1}$ and the ...
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### Fibration with CW-complex as basespace admits retraction

Suppose $f:E\rightarrow B$ has the right lifting property with respect to all CW-pairs $(X,A)$. Then $f$ is a Serre fibration and also a weak-homotopy equivalence. But want i want to prove is the ...
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### Homotopy of maps $(D^n, S^{n-1}) \longrightarrow (X,A)$ relative $S^{n-1}$

Suppose that for a map $f: (D^n, S^{n-1}) \to (X,A)$ (where $(X,A)$ is an arbitrary pair of spaces) there exists a homotopy $H: D^n \times I \to X$ with $H(\_,0)=f$, $H(s,t) \in A$ for $s \in S^{n-1}$ ...
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### Retraction and intersection

Let $X$ be a topological space, and consider two open subsets $U$, $V$ of $X$ such that there exist two continuous maps $r_{U}: X\longrightarrow U$, $r_{V}:X\longrightarrow V$ which are homotopically ...
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### Fiber sequence of principal bundles

Let $G$ be a group, either a Lie group or a discrete group. Let a principal $G$-bundle $$G\to E\to B,$$ then $B=E/G$, the orbit space under action of $G$. Let $BG$ be the classifying space of $G$. ...
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### Does the 2-functor $PsAlg\to \mathfrak{X}$ reflect equivalences?

Consider a $2$-monad $T: \mathfrak{X}\to \mathfrak{X}$ and consider its 2-category of pseudoalgebras $PsAlg$. There is a forgetful functor $U: PsAlg\to \mathfrak{X}$. Does this forgetful functor ...
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### Hopf invariant and homotopy groups of spheres

I am trying to understand how to use Hopf invariant, to calculate $\pi_{4n-1}(S^{2n})$. I've started with defining a new space $X$ adjoining $D^{4n}$ to $S^{2n}$ via a map $\phi\in\pi_{4n-1}(S^{2n})$...
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### Reference for secondary cohomology operations

I am learning some homotopy theory and am currently reading Mosher and Tangora. I love the content of this book, it's very terse and comes straight to the point. At the same time I find it very ...
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### Is this morphism of spectra zero in the stable homotopy category?

Let $f\colon A\to B$ a morphism of spectra and suppose that both spectra $A$ and $B$ have only one non-zero stable homotopy group $\pi_n$, more precisely $$\pi_k(A)=0=\pi_k(B)$$ for $k\neq n$. ...
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### null homotopy of 1-skeleton of simply connected simplicial complex induced by homotopy identity

I am reading a book suggesting that the following is true: Let $X$ be a simply connected (maybe finite) simplicial complex. Then there exists a continous map $$g : X \to X,$$ homotopic to the ...
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### mapping cone and cylinder

Given a map of spaces $f:X \to Y$, the mapping cylinder is the adjunction space $$cyl(f)=(X \times [0,1]) \cup_f Y$$ where we regard $f$ as a map $f: X \times \{1\} \to Y$.\ On the other hand the ...
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### What is meant by saying that two paths in $X$ from $x_0$ to $x_1$ are equivalent?

Suppose that X is a topological space and $x_0$, $x_1$ are points of $X$. What is meant by saying that two paths in $X$ from $x_0$ to $x_1$ are equivalent? I presume it's enough to just say the path ...
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### Why a convergent succesion does not have the same homotopy type of a CW-Complex?

The question is pretty much in the title; If my space is $\{1/n\}_{n\in \mathbb{N}} \cup \{0\}$ why it isn't homotopically equivalent to a CW-Complex?
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### Show that $\mathbb{R} P^3$ is not homotopy equivalent to $\mathbb{R} P^2 \vee S^3$.

I'm studying for an oral qualifying exam in algebraic topology, going through questions in various tests published on the interwebs. Here's a rather straightforward question from this exam that is ...
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### Moduli spaces, stacks and homotopy theory

For my final-year project (not this academic year but next) I'm hoping to write a relatively complete account of the basic theory of schemes used in modern algebraic geometry. My supervisor thinks ...
I want to prove the following statement: Let $(X, x_0)$ be a pointed space, and let $X' = X\cup_{\alpha} e^{n+1}$ be obtained from $X$ by adjoining an $(n + 1)$-cell. Then the inclusion \$i : X\...