# Tagged Questions

Questions about algebraic methods and invariants to study and classify topological spaces: homotopy groups, (co)-homology groups, fundamental groups, covering spaces, and beyond.

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### deformation-retracts into a point / contractible : what is the difference?

Okay, I keep reading the definitions over and over, but I don't see the difference between the two ; apparently spaces that deformation retract onto a point are contractible, but the opposite is not ...
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### What is the fundamental group of $X=S^2\cup\text{a disc parallel to the plane of equator}$?

How I imagine it: I think a cell structure of $X$ consists of 1 0-cell, 1 1-cell (the equator)-call it $a$, 3 2-cells (north-south hemispheres and the disc). But I'm not sure about the 2-cells, ...
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### The inclusion map $V\subset U$ is nullhomotopic. - Is this proof legit?

There are many discussions has done on Hatcher 0.5, some here, and some there. But I am primarily uncertain about the invoke of tube lemma. Wonder if someone would kind help me take a look at my proof ...
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### Cohomology of complex projective space

Hello : I would like to know how to compute the cohomology of complexe projective space : $H^p ( \mathbb{P}^n ( \mathbb{C} ) , \mathbb{Z} )$. Thanks a lot.
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### Construct an explicit deformation retraction of $\mathbb{R}^n - \{0\}$ onto $S^{n−1}$.

Just getting started with Hatcher, wondering if I get the right idea at the beginning? Problem 0.2, Page 18: Construct an explicit deformation retraction of $\mathbb{R}^n - \{0\}$ onto $S^{n−1}$. ...
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### Turning higher spheres inside out

I know that one can turn a sphere $S^2$ in $\mathbb{R}^3$ inside-out having at each time an immersion of $S^2$ into $\mathbb{R}^3$. It is called Smale paradox. There is beautiful animation about that. ...
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### constructing a genus 2 surface from 8-gon

I am requesting some help or reference for visualization? I am having a hard time constructing a genus 2 surface from 8-gon. May I request for some reference? Here's the construction I used from ...
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### What's the classification of CW complexes formed by gluing a 2-cell to a circle?

After this answer, the following question comes : What's the classification (up to homeo.) of CW complexes formed by gluing a 2-cell to a circle ?
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### Is a CW complex, homeomorphic to a regular CW complex?

Is a CW complex, homeomorphic to a regular CW complex ? Regular means that the attaching maps are homeomorphisms (1-1). In particular, an open (resp. closed) $n$-cell, is homeomorphic to an open (...
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### Is a uniquely geodesic space contractible? I

Is a uniquely geodesic space contractible ? We assume in addition that closed metric balls are compact. A post without this extra assumption is here.
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### Group Extension and Classifying Space

If $$0 \to H \to G \to G/H \to 0\$$ is a group extension, under what conditions do we have a fibration of the form $$BH \to BG \to B(G/H),$$ where $BG$ is a classifying space of $G$? Suppose ...
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### Deformation retract needs to be smooth?

So I am not quite sure that why none of these three is a deformation retract - is that because of the corners? But I don't remember deformation retract rely on smooth criteria, instead, on continuous ...
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### Hatcher's Algebraic Topology Chapter 0 at Page 3

I have a question from Hatcher's Algebraic Topology Chapter 0 at Page 3: http://www.math.cornell.edu/~hatcher/AT/ATch0.pdf One could equally well regard a retraction as a map $X\to A$ restricting ...
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### Why does the letter $X$ retract onto a point? (Hatcher's Algebraic Topology, Chapter 0, pg 2)

I am stumble at a statement from Hatcher's Algebraic Topology Chapter 0 at Page 2 which can be found here. The thick $\mathbf{X}$ deformation retracts to the thin $X$, which in turn deformation ...
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### Find the Fundamental group of this space

Let X be the space obtained from $S^2$ by identifying (x; y; 0) with (-x; -y; 0), for all (x; y; 0)$\in$ S2. Compute $\pi_(X).$. I know to choose the open sets so that they each deformation retract ...
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### Uniquely geodesic spaces

The purpose of this list issue is to better understand the class of uniquely geodesic spaces. I'm looking for two different things : Overclass : for example geodesic space or contractible space. ...
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### Dimension of the homology group with coefficients in $\mathbb{Z}/2\mathbb{Z}$.

Charles Weibel writes in his survey of homological algebra Riemann de fined a surface $S$ to be $(n + 1)$-fold connected if there exists a family $C$ of $n$ closed curves $C_j$ on $S$ such that ...
Proposition Any path-connected CW-complex is homotopy equivalent to a CW-complex with precisely one 0-cell. Proof (Sketch) Let $X$ be a path-connected CW-complex, so $sk_1(X)$ is a connected graph. ...