# Questions tagged [retraction]

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### Continuity of a weak deformation retraction of the zigzag comb space

This quesiton is related to Why does the "zig-zag comb" weakly deformation retract onto a point? and How to prove "zigzag comb" is contractible (Hatcher ex. 0.6(b)). In both links, ...
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### Subspace of $[0,1] \times [0,1]$ that deformation retracts to any point in $[0,1] \times \{0\}$, but not any other point.

This is an exercise that I'm doing, and I would like some comment on my solution. This is the picture I have in my head, where $(p,0)$ is the point $X$ deformation retracts to, and $(x_1,x_2)$ ...
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### Are there any $\mathbf{Z}[x]$-algebra maps of the following form?

An exercise I am computing reduces to the following question: Is there a map $f$ of $\mathbf{Z}[x]$-algebras such that the composition  \mathbf{Z}[x,x^{-1}]\xrightarrow{f}\mathbf{Z}[x]\to\...
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### A counter-example showing the inclusion $\{x_{0}\}\hookrightarrow X$ is a homotopy equivalence but $\{x_{0}\}$ is not a deformation retract of X

This is the exercise from Munkres' section 58 Exercise 8. Find a space $X$ and a point $x_{0}$ of $X$ such that the inclusion $\{x_{0}\}\hookrightarrow X$ is a homotopy equivalence but $\{x_{0}\}$ ...
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### A compact Riemann surface with boundary strongly deformation retracting onto its boundary

Let $X$ be an open Riemann surface and $K$ an open, relatively compact, connected subset of $X$. It seems to me that it is impossible for $\overline{K}$ to strongly deformation retract onto its ...
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### Does there exist two h-equal but not h-equivalent topological spaces?

In his "Theory of retracts", Borsuk defines the concept of h-dominance (h is for homotopically), that is: given two (in this post i consider only Hausdorff spaces) topological spaces $X,Y$ we say that ...
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### deformation retraction of the torus with 1 pt. deleted onto 2 circles intersecting at a pt. and that of $\mathbb R^n \setminus \{0\}$ onto $S^{n-1}$.

Here are two questions with their answers: My question is: Why in the second term of $H(x,t)$, in the first problem we have the term $t (g^{-1} \circ f)$ while, in the second term of $H(x,t)$...
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### A small discussion about Hatcher Q.1, chapter 0

The question and its solution are given below: My question is: Should the deleted point of the torus be the intersection point of the two circles? If yes why? and if no, why also?
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### Changing the codomain in the definition of the retraction map.

In Allen Hatcher, Algebraic topology, the following paragraph exists: And I am wondering if in the sentence starting with "From a more formal viewpoint .....", I changed the codomain of the ...
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### Deformation retraction arising from a mapping cylinder.

In Allen Hatcher Algebraic topology, chapter zero, I had the following: My questions are: 1- Why the mapping cylinder is the quotient space, what makes it specifically a quotient space? 2-...
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### Are open local embeddings equivalent to local diffeomorphisms? (Do not use immersions)

Open immersions are equivalent to local diffeomorphisms, and immersions are equivalent to local embeddings, so obviously yes. I would like to understand why open local embeddings equivalent to local ...
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### $\mathbb{S}^n$ without two points

In "An Introduction to Algebraic Topology" of Rotman, Exercise 1.31 asks to show that the equator $\mathbb{S}^{n-1}$ is a deformation retract of $\mathbb{S}^n\setminus\{a, b\}$. I thought that if ...
Does the word "subspace" here imply "linear subspace"? A subspace $Y$ of $\mathbb{R}$ is said to be a retract of $\mathbb{R}$ if there exists a continuous map $r: \mathbb{R} \to Y$ such that $r(y)= ... 2answers 291 views ### Why is$S^1\times \{1\}$homotopy equivalent to the solid torus$T^2 = D^2 \times S^1$? (see attached picture) I am currently self-studying the basics of algebraic topology and i just learned the definitions of retract, deformationretract and homotopy equivalence. Now in my book there is an example of a ... 1answer 47 views ### If$A$has a neighbourhood in$X$that deformation retracts onto$A$then does$(X,A)$have HEP? Let$(X,A)$be a topological pair. Assume that$A$has a neighbourhood$V$such that$V$deformation retracts onto$A$. Then can we say that$(X,A)$has the homotopy extension property? Edit: I know ... 2answers 83 views ### Suppose$A \subset V$. How does a deformation retraction of$V$onto$A$induces a deformation retraction of$V/A$onto$A/A$? Suppose$A \subset V$. If there is a deformation of$V$onto$A$, then there exists maps$i: A \hookrightarrow V$and$r: V \to A$such that$ri=Id_A$and$ir \simeq Id_V$. How does this ... 1answer 140 views ### Fundamental group as a product of normal subgroups Let$A$be a retract of a non-empty topological space$X$and let$a \in A$. Let's denote$r : X → A$the retraction and$i : A → X$the inclusion. prove that$i_∗ (π_1 (A, a))\triangleleft π_1 (X, ...
Suppose $M$ is a connected, noncompact 2-manifold, and its boundary $\partial M$ is a circle. What's the simplest way to show there is a retraction $r: M\rightarrow \partial M$? Here are some ...