# Tagged Questions

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### $S^{n-1}$ is not a deformation retract of $\mathbb{P}^n(\mathbb{R})/ B(0,1)$.

Let $n$ be $\geq 2$, $$\mathbb{P}^n(\mathbb{R}) \supset S^{n-1}= \lbrace [1,x_1,...,x_n] | x_1^2+...+x_n^2=1 \rbrace$$ and $B(0,1)= \lbrace [1,x_1,...,x_n] | x_1^2+...x_n^2<1\rbrace$. Show that ...
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### cells of quotient CW complex

Let $X$ be a CW complex and $Y$ a CW subcomplex. If $X$ has no cell of dimension $n$, for some $n>0$, then $X/Y$ has no cell of dimension $n$. Is it true? Why?
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### $X$ is contractible and $Y$ is path connected then $[X,Y]$ has a single point

Im trying to show that: for $X,Y$ topological spaces $X$ is contractible and $Y$ is path connected then $[X,Y]$ has a single point while $[X,Y]$ denote the set of homotopy classes of maps of $X$ ...
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### Motivation for the proof of the associativity of multiplication of equivalence classes of paths

After having defined the equivalence classes of paths in a topological space in chapter two of the book A Basic Course in Algebraic Topology, William S. Massey proves the lemma The multiplication ...
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### The cone minus its apex deformation retracts onto its basis

Let $X$ be a topological space and $$C(X)=X\times [0,1]/X\times \{0\}$$ be the cone on $X$. Call $P$ the apex of the cone. I want to show that $C(X)-P$ deformation retracts onto $X\times \{1\}$. My ...
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### Good pair vs. cofibration

It can be shown that $i:A\hookrightarrow X$ is a closed cofibration if and only if there is a map $\varphi:X\to I=[0,1]$ and a homotopy $H:U\times I\to X$ on some neighborhood $U$ of $A$ such that ...
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### Stability of the nonempty intersection of an open set $A$ with a set $S$ under homotopy?

To be more precise: let $F(x,t) : R^2 \times I \to R^2$ be a homotopy of open maps $F(_,t)(x)$ (the restriction of $F$ to some fixed $t$) (the homotopy is continuous in both variables). Suppose that ...
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### What's the calculation formula of topological number for mappings of $\pi_{3}(S^2)=\mathbb{Z}$?

It is well-known that, when mapping $|\vec{n}(\vec{x})|=1$, we can use $N=\int{\mathrm{d}x_1\mathrm{d}x_2\vec{n}\cdot(\partial_1\vec{n}\times\partial_2\vec{n})}$ to calculate the topological winding ...
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### The unit ball in $\mathbb{R}^{n}$ and a point are homotopically equivalent.

I will appreciate if someone could explain to me the solution of the following problem: The unit ball in $\mathbb{R}^{n}$ and a point are homotopically equivalent. Def 1: Two spaces $X$, $Y$ are ...
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### How do I prove that $U(r) \to S(r,n) \to G(r,n)$ is a fibration?

$U(r)$ here is unitary group of $\mathbb{C}^n$, $S(r,n)$ is the Stiefel manifold of $r$-frames in $\mathbb{C}^n$ and $G(r,n)$ is the Grassmannian manifold of $r$-planes in $\mathbb{C}^n$. I've tried a ...
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### Homeomorphic or Homotopic

Q1: Are the Fig (a) and (b), the equivalence "=" is Homeomorphic or Homotopic? ps. for details of the figures see Ref here. I learned that "The characterization of a homeomorphism often ...
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### Topology of space of symmetric matrices with fixed number of positive and negative eigenvalues

Let $M$ be real non-singular symmetric $n \times n$ matrix with $p$ positive and $n-p$ negative eigenvalues. What is the topology of the space of such matrices? For a trivial case $n=1$ the matrix is ...
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### homotopical equivalence of projective real space less a line

let $r$ a projective line of the projective real space. How can i prove that $\mathbb{P} ^3(\mathbb{R}) - r$ is homotopical equivalent to $S^1$?
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### Foundamental group of $n+1$ spheres in $\mathbb{R}^{n+1}$ that touch two by two

How can i calculate the foundamental group of three $S^2$ in $\mathbb{R}^3$ that touches two by two in one point (if you take any two spheres, they touch only in one point) ? I know that is ...
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### Homotopic maps between spheres

I have read somewhere that two maps $f,g:S^n\rightarrow S^n$ satisfying $$|f(x)-g(x)|<2 \qquad \forall \ x\in S^n$$ are homotopic. How can one show this (or does someone have a reference)? I ...
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### Is there any standard terminology for the quotient of a topological group by the connected component of the identity?

If $G$ is any topological group, then the connected component of its identity is a closed normal subgroup $H$. It follows that $G/H$ is a totally disconnected topological group. Often, $G$ will be ...
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### Example for a space that is contractible to precisely one of its points

Give an example for a space that is contractible to one of its points and is not contractible to another of its points. I am really curious about that space, I have thought about tree with $n$ ...
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### Quasi circle is not contractible

I'm trying to show that the quasi circle (picture below) doesn't have the homotopy type of a CW complex. I proved that all homotopy groups are zero. Now I need to show that it is not contractible to ...
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### How to show that homotopy is preserved after composition?

I have two homotopies: $f\simeq f'$ and $y\simeq y'$. How can I show that $fy\simeq f'y'$ is again a homotopy?
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### mapping homotopic to the identity map

Please give me a hand with this problem, It was on my exam, and I just couldn't solve it. Suppose $\phi:\mathbb{S}^2\to\mathbb{S}^2$ is a mapping, homotopic to the identity map. Show that there is ...
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### About the universal bundle $EG\rightarrow BG$

For a topological group $G$, we define $EG$ to be the infinite join of $G$, and $B$ to be the quotient of $EG$ by the left action of $G$. Explicitly $EG$ can be expressed, as a set, as ...
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### Proof that two homotopy inverses are homotopic

Let $X$ and $Y$ be topological spaces. A continuous mapping $f : X \to Y$ is said to be a homotopy equivalence if there exists $g : Y \to X$ continuous such that $g\circ f$ is homotopic to $id_{X}$ ...