Questions tagged [cw-complexes]

For questions about CW complexes (topological spaces which are built up using balls of varying dimensions known as cells).

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Path component of a CW-complex is a subcomplex.

Let $X$ be a CW-complex. I want to prove each of its path components contains a zero cell. For that, I want to prove that each of its components is again a CW-complex. Please give me some hints.
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50 views

Contractibility of CW complex without Whitehead

Suppose I have a CW complex $X$ with skeleta $(X_n)_{n\ge 0}$ such that $\pi_k(X)=0$ for all $k\ge 0$. I want to conclude that $X$ is contractible without invoking Whitehead's theorem. It would be ...
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49 views

CW complex with one 0-cell [closed]

Is there any example for CW complex with one 0-cell that is not connected?I don't think it's possible.
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84 views

Attaching 1-cells to CW-complex affects homotopy groups?

Is it true that attaching 1-cells to a CW-complex doesn‘t change it’s higher homotopy groups $\pi_n$ for $n\ge 2$? (I am aware that a corresponding result for cells of higher dimension is far from ...
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1answer
222 views

Homotopic maps and attaching spaces

Let $f, g : \mathbb{S}^{n-1} \to X$ be two continuous maps from the sphere into a compact and Hausdorff space $X$. I want to show that if $f$ and $g$ are homotopic, then attaching an $n$-cell to $X$ ...
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48 views

CW approximation of a topological group

Suppose we have a topologiacal group $G$. How do we construct a CW-complex $\bar{G}$ which is also a topological group, such that $\bar{G}\to G$ is a weak equivalence?
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312 views

Wedge sum of spheres is the quotient $X^n/X^{n-1}$

As in the title, I want to prove that $\bigvee_jS_j^n=X^n/X^{n-1};\ X$ is a $CW$ complex and $X^n$ and $X^{n-1}$ are the $n-$ and $n-1$-skeleta. Below, I present a sketch of an attempt using pushouts, ...
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1answer
111 views

Bruhat-Tits Building of $PGL_3$: What does it look like?

I am very new to this topic, and my background is mostly in graph theory and basic algebra. What I want for now is to understand the structure of the dimension $2$ complex $\mathcal{B}(PGL_3(K))$ ...
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1answer
298 views

CW-complex structure on the quotient

Let $X$ be an $n$-dimensional CW-complex and let $A \subseteq X$ be a subcomplex. I want to show that the quotient space $X/A$ admits a structure of a CW-complex with skeletons $(X/A)^j := \pi(X^j \...
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1answer
129 views

Is a bouquet of circles always a CW complex?

Let $X$ be a cell complex with one $0$-cell and an arbitrary (possibly infinite of any degree) number of $1$-cells which are self-loops to the only $0$-cell. Clearly $X$ is closure-finite (property C ...
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1answer
90 views

Realizing a CW Complex as an Adjunction Space: Munkres' Proof

Suppose $Y$ is a $CW$ complex, of dimension $p-1,\ \sum B_{\alpha}$ is a topological sum of closed $p-$ balls. Then, if $g:\sum \partial B_{\alpha}\to Y$ is a continuous map, the adjunction space $X=Y\...
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114 views

closed cells form a covering of the CW complex.

A CW complex $X$ is the union of open cells $\{e^i_\alpha\}$ and these open cells are disjoint subsets of $X$, so these open cells form a partition of $X$. Now if we take the closed cells $\{\bar e^...
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How to introduce a CW structure on RP^n?

My first course in topology is going extremely fast, and does not seem like rigorous mathematics. Last lecture, we were given the definition of CW-structures, but did not do any examples. Yet we were ...
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301 views

An open set in a CW complex

I'm interested in the definition that says that a CW complex is defined as a partition of open cells $\{e^i_{\alpha}\}$ verifying "some conditions"(see Munkres page 214). We endow $X$ with a ...
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1answer
173 views

CW structure of the universal cover

Consider the universal covering $p:\tilde X\to X$. Given a cw structure on $X$, I want to understand how to find the cw structure of $\tilde X$. Let's take an example $$p:\mathbb R\to S^1;\;\;t\...
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1answer
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Skeleton of CW product $\Sigma^i X \times \Sigma^i Y$

I'm in trouble with an assertion in Hatcher's algebraic topology; let $X, Y$ be CW-complexes, and let's denote with $\Sigma$ the reduced suspension (it is the usual suspension $SX=X \times[0,1]/X \...
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1answer
157 views

Can we define the Euler characteristic of an infinite-dimensional CW complex?

For a finite CW complex $X$, the Euler characteristic $\chi(X)$ is defined to be the alternating sum $\sum_k(-1)^kc_k$ where $c_k$ is the number of $k$-cells of $X$. Note, the Euler characteristic of $...
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2answers
108 views

Weak homotopy equivalence between $\Omega \underset{\rightarrow}{\lim} \ Z_n$ and $\underset{\rightarrow}{\lim} \ \Omega Z_n$

Let $Z_1 \rightarrow Z_2 \rightarrow\cdots$ be an arbitrary sequence of CW-complexes and let $\Omega X$ denote the loop space over $X$. In Allen Hatcher's "Algebraic Topology" (http://pi.math.cornell....
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1answer
38 views

$(X \wedge K)/(A \wedge K)= (X/A) \wedge K$

I have to prove that the following equality $$(X \wedge K)/(A \wedge K)= (X/A) \wedge K$$ holds for a CW-pair $(X, A)$ and a fixed CW-complex $K$; here $\wedge$ denotes the smash product, defined as $...
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59 views

Every cell complex is homotopy equivalent to a CW complex

Every cell complex is homotopy equivalent to a CW complex. A cell complex is more general than a CW complex in that the cells need not be attached in increasing order of dimension. Any hints on ...
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1answer
69 views

Smooth cell in a CW complex

Suppose a $k$-cell in a CW complex is given a smooth structure, so that it is also a smooth $k$-manifold. Is that cell diffeomorphic to the open $k$-ball?
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48 views

In a CW complex, what is the closure of a cell $e_{\alpha}^n$?

Is the closure of a cell $e_{\alpha}^n$ exactly $\Phi_{\alpha}^n(D_{\alpha}^n)$?I can only show $\Phi_{\alpha}^n(D_{\alpha}^n)\subset\overline{e_{\alpha}^n}$.
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2answers
249 views

What is the cellular homology of a CW complex $X$ from long exact sequence of relative homology?

Consider the handle decomposition of the manifold $Y:=Y_N$ where we let $Y_k=Y_{k-1}\cup_{\chi} H^{\gamma_k}$ be a $\dim Y_{k-1}$-manifold with a $\gamma_k$-handle attached $H^{\gamma_k}=D^{\gamma_k}\...
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1answer
264 views

Show that any cell complex is homotopy equivalent to a CW complex

We will give the definitions of cell complex and CW complex we use at the end of the post. Briefly speaking, "in a cell complex you don't have to glue cells in the order of their dimension, whereas ...
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1answer
126 views

Attaching maps of products of CW-complexes.

I am a bit confused about how the "product attaching-maps" work. For instance, if I wanted to find the cell structure for $S^1\times S^1$ then I can proceed similarly to here. So we will call $e_0$ ...
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1answer
65 views

Identifying 3 sides of a polygon.

When identifying opposite sides of polygons we obtain a surface. For instance, the torus can be obtained by gluing together opposite sides of a square. I saw a claim that if we identify 3 sides of a ...
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1answer
110 views

The character group is an algebraic group.

Let $X$ be a finite dimensional connected CW complex. Let $G=\pi_1(X)$. Let $\hat G=Hom(G,\mathbb C^*)$ the character group of $G$. Then $\hat G$ is an affine algebraic group. Could someone explain ...
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474 views

Using the constructive definition of a CW-complex to prove that it is Hausdorff.

In class, I was given a constructive definition of a CW-complex and told that it's an easy exercise to prove that it is Hausdorff. I included the definition given in class below; We set $X^0 = \...
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141 views

Map induced in homology by inclusion

Using Mayer-Vietoris sequence often one needs to know which map is induced in homology by an inclusion, let me try to explain with an example: Let's take the real projective plane with a hole i.e. $X=...
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Eilenberg-Zilber Theorem Proof

Let $X, Y$ be CW complexes and the following proof (from J. P. May's "A Concise Course in Algebraic Topology"; see page 102. Here the full pdf document: https://www.maths.ed.ac.uk/~v1ranick/papers/...
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Restriction of Attatching Map for CW Complexes Closed

A CW complex $X$ is a Hausdorff space and can be interpreted as a colimit $X = colim_k X_k$ of cells $X_k $ which satisfy can inclusion relations $X_{k-1} \subset X_k$ for each $k$ and are defined ...
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1answer
106 views

The space obtained by attaching a subspace of a topological space to itself.

Just asking a special case of attaching, it seems pretty vague to me. Let $Y$ be a space and $X$ be its subspace. Then the space obtained by $Y$ attaching its subspace $X$ via $\iota$ (the embedding) ...
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1answer
160 views

Simplicial approximation to CW Complex

In Hatcher's book, the following theorem is mentioned (Th 2C.5, page 182): Every CW complex $X$ is homotopy equivalent to a simplicial complex, which can be chosen to be of the same dimension as $X$...
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1answer
108 views

If a graph homomorphism induces an isomorphism, is it a homotopy equivalence?

I am taking graphs to potentially have loops and multiple edges. That is, they look like 1-dimensional CW complexes. If $f\colon\Gamma\to\Delta$ is a graph homomorphism (a.k.a. graph map, map of ...
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1answer
307 views

Example of a CW complex that is not a $\Delta$-complex?

Hatcher notes in chapter 2.1 (in the paragraph just preceding the section on simplicial homology (page 104 in my edition)), that all $\Delta$-complexes can be realized as CW complexes with some added ...
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142 views

$Mat_{ \infty,n }(\mathbb C)$ of max rank is contractible

We want to show that for a given $n\in\mathbb{N}$, the set of all the $\infty\times n$ matrices $Mat_{\infty,n}(\mathbb{C})$ of max rank $n$ form a contractible space. This set is obtained by taking ...
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238 views

Associativity of join for CW-complexes

I am currently self-studying a course in algebraic topology and one of the problems I encountered is to prove that the join operation defined as $$X \ast Y=X\times Y\times I/(x_1,y,1)\sim(x_2,y,1), (x,...
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1answer
296 views

Intersection of subcomplexes

Ok, so intuitively it's clear that the intersection of two subcomplexes of a CW-complex should be a subcomplex as well, but reading the inductive definition of a CW-complex, nowhere does it say that a ...
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55 views

Is taking adjunction space compatible with topological product?

I am reading John W. Milnor, Morse Theory, though it might be a bit difficult, and I was showing in Lemma 3.6 that $k\circ l$ is homotopic to identity map. (Until here is for those who have the ...
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Constructing a regular cell-decomposition of $\mathbb{S}^n$

Okay so here are a few definitions first. Definitions: If $X$ is a nonempty topological spaces, a cell decomposition of $X$ is a partition $\Gamma$ of $X$ into subspaces that are open cells of ...
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2answers
131 views

Whitehead theorem

If a map $f:X\to Y$ of CW-complexes induces an isomorphism on homotopy groups it is a homotopy equivalence by the Whitehead theorem. In particular it also induces isomorphisms on homology and ...
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1answer
219 views

Reference for CW-Complexes

I am taking a class on algebraic topology, and this concept was introduced quite quickly: it was the subject of a week's lecture and then we just kept on using it during the exercises. And now I still ...
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1answer
324 views

Is CW approximation interesting?

Each topological space $X$ is weakly equivalent to a CW-complex $X'$. That is, there exits a map $f:X'\rightarrow X$ that induces isomorphisms on $\pi_n$ for each $n\in \mathbb N$. My question is ...
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70 views

Weak product of Eilemberg MacLane spaces

I'm studying some homotopy theory of topological monoids from the book Algebraic Topology from a Homotopical Viewpoint. I'm trying to understand the corollary below. I'm stuck on the first claim of ...
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1answer
144 views

colimit of CW-complexes

For any $n\geq 1$ let $(X_n,X_{n-1})$ be a CW-pair, that is we assume that $X_n$ is a CW-complex and $X_{n-1}$ a subcomplex. I'm trying to show that $X:=\mathsf{colim} \, X_n$ is a CW-complex. Any ...
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2answers
233 views

Homology groups of $(S^2\times S^2)\cup_{\Delta} D^3$

We define a space $X$ by $$X=(S^2\times S^2)\cup_{\Delta} D^3$$ where $S^2$ is the $2$-sphere, $D^3$ is the $3$-disk, and $\Delta\colon S^2\to S^2\times S^2$ is the diagonal map, so we attach a $3$-...
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258 views

CW complex is connected iff 1-skeleton is connected

Let $X$ be CW-complex and by $X^n$ I denote its n-skeleton. Here is my attempt to prove that connectedness of $X$ and $X^1$ are equivalent. First, assume that $X^1$ is not connected, which means one ...
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1answer
54 views

Spaces homotopy dominated by a Moore space

‎A Moore space of degree $n\geq 2$ is a simply connected $CW$-complex X with a single non-vanishing homology group of degree $n$‎, ‎that is $\tilde{H}_{i}(X,\mathbb{Z})=0$ for $i\neq n$‎. ‎We write $X=...
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69 views

Using Homotopy Extension Propperty to prove that $S^{\infty}$ is contractible

I know that the question: ''Is $S^{\infty}$ contractible?'' has already been asked, but none of those post seem to answer it the way I need (Maybe they do, and my lack of knowledge prevents me from ...
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1answer
72 views

CW Complexes $X/B$ and $X/A \cap B$ are homeomorphic

Let $(X,A,B)$ be a cellular triad, therefore $X,A,B$ are CW complexes with $X = A \cup B$ (especially $A,B$ are sub complexes of X$. How to show that $X/B$ is homeomorphic to $X/A \cap B = A \cup B/A ...

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