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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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CW Structure on Self-Homeomorphism Groups

If $X$ is a finite CW complex then according to Milnor's article On Spaces Having the Homotopy Type of a CW-Complex, the mapping space $$Map(X,X)$$ (which we furnish with the compact-open topology) ...
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Definition of mapping telescope

In Kochman's stable homotopy theory, pg 121 prop 4.24 We let $X$ be a based CW complex. Let $X^n$ be an increasing sequence of subcomplex whose union equals $X$. We define $$TX = \bigcup_{n \ge ...
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Prerequisite of understanding a topological construction in Spectral Sequences

This is a post on the construction of a spectral sequence. I am in fact lost in the first paragraph. Let $B$ be a CW complex and $\pi\colon X\to B$ a Serre fibration. Put $X^k=\pi^{-1}(B^k)$. A ...
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same CW chain complex implies homotopic?

Is it true that if two spaces X,Y have the same CW chain complex i.e. ${C^{CW}(X)}_n={C^{CW}(Y)}_n$ and also the same differential maps $d_n$ at each stage, then $X \simeq Y$ ? If the CW chain ...
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Confusion regarding CW complex structure of product

I find that if $X,Y $ are CW complexes then $X \times Y$ is a CW complex with skeleta $(X \times Y)^{(n)}=\cup_{p+q=n} X^{(p)} \times Y^{(q)}$ . So while considering $S^n$ with CW complex structure: ...
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Mistake in computing boundary maps in $S^n$

I am giving $S^n$ the CW complex structure with $2$ $k-$cells in each dimension $0\leq k\leq n$ . The attaching maps are given by the pushout diagram. $\require{AMScd} \begin{CD} S^{k-1}\cup S^{k-1} ...
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CW complexes are the cofibrant objects in the Quillen model structure on Top?

If $J$ is a class of maps in a category, the $J$-cellular maps are by definition transfinite compositions of pushouts of coproducts of maps in $J$. Now if $J$ denotes the family of inclusions $S^{n-...
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bounded generalized homology theory : isomorphism

Let $h_*$ be a generalized homology theory. Write $X = \bigcup_{s \ge 0} X^s$, where $X^s$ is skeletal filtration of CW complex $X$. It is claimed in Kochman's pg124, line 4 that: if $P$ is the ...
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Coverings of CW complexes are also CW complexes: How do I show that it has the weak topology?

Let $X$ be a CW complex, $p:E\to X$ a covering map. Then $E$ has an induced CW complex structure defined as follows. If $\Phi:D^n\to X$ is a covering, it lifts to a map $D^n\to E$ (since $D^n$ is ...
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Deriving Kunneth formula for Cellular Homology

Let $X$ and $Y$ be finite CW complexes. We equip $X\times Y$ with the usual CW complex structure. Thus $C^{CW}_*(X \times Y) \cong C^{CW}_* (X) \otimes C^{CW}_*(Y)$ as chain complexes. I know for ...
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Characteristic functions in a CW complex are related by a homeomorphism

The book "The Topology of CW Complexes" mentions the following result in passing, but does not provide a proof. Let $(X, \mathcal{C})$ be a CW complex, $B_k \subset \mathbb{R}^k$ be the origin-...
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Quotient of $n$-th skeleton by smaller dimensional skeleton

Suppose $X$ be a CW complex. Then we know that, $$\frac{X^n}{X^{n-1}}=\underset{\text{$\sigma$ an $n$-cell}}{\large\lor} \Bbb S^n,$$ Where $X^n$ is $n$-th skeleton of $X$ and $X^{n-1}$ is $(n-1)$-th ...
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Is a union of closures of cells in a CW complex closed?

Let $X$ be a CW complex with cell-partition $\mathcal{C} \subset \mathcal{P}(X)$, and $\mathcal{D} \subset \mathcal{C}$. Is $A = \bigcup \{\overline{D}(X) : D \in \mathcal{D}\}$ closed in $X$, where $\...
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Hatcher Exercise 3.2.16

Hatcehr Exercise 3.2.16. Show that if $X$ and $Y$ are finite CW complexes such that $H^∗(X;\mathbb Z)$ and $H^∗(Y;\mathbb Z)$ contain no elements of order a power of a given prime $p$, then the ...
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example of a topological space which is Normal, locally contractible, semi locally simply connected but not a CW complex.

We know that if a topological space $X$ is a CW complex then it is normal, locally path-connected,semi-locally simply connected. And using these properties we can conclude that a space is not a CW ...
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$H_n(X, F_p) = 0$ implies $H_n (X,Q)=0$ for certain CW-complexes $X$

Consider a CW-complex $X$ with exactly one cell in each dimension. Suppose there is a prime $p$ such that $H_n(X, F_p) = 0$ for all $n\geq 1$. How can I show that this implies $H_n (X,Q)=0$ for all ...
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Fundamental group of tetrahedron

Consider the following double tetrahedron We glue $DCE$ to $CBA$, $CBE$ to $BDA$ and $BDE$ to $DCA$. We call the resulting space $L$. I want to find a cell-structure on $L$ with only two $0$-cells ...
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Cellular homology of suspension

As part of an assignment I would like to show directly from the definition of cellular homology that if $X$ is a CW-complex then: $$H^{\mathrm{Cell}}_{n+1}(S(X);\mathbb{Z})\cong H^{\mathrm{Cell}}_n(X;\...
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Is a closed embedding of CW-complexes a cofibration?

It is a standard fact that the inclusion of a sub-CW-complex into a CW-complex is a cofibration, it follows from the fact that the inclusions $S^k\to D^{k+1}$ are, and that they are preserved by ...
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Construction of Moore Space

While reading the construction of Moore space from Hatcher's Algebraic topology on page 143 , I faced the following problem:--- Let $G$ be an abelian group and $0\rightarrow K\rightarrow F\rightarrow ...
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Let $X$ be a connected CW complex and $G$ a group such that every $\pi_1(X)\to G$ is trivial. Show that every $X\to K(G, 1)$ is nullhomotopic.

Question 2 in Chapter 1.B in Hatcher's Algebraic Topology: Let $X$ be a connected CW complex and $G$ a group such that every homomorphism $\pi_1(X)\to G$ is trivial. Show that every map $X\to K(G, ...
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Homology of cube with some identifications.

I have problems solving this problem. I know we can use a CW complex made of 1 0-cell, 3 1-cells, 3 2-cells and one 3-cell, but i really can't compute degrees... Thanks for any help :')
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Hatcher Exercise 1.2.8

I am trying to prove the following excersise (1.2.8) from Hatcher's Algebraic Topology: Given 2 tori $S^1\times S^1$ and identifying $S^1\times \{x_0\}$} compute the fundamental group. My approach is ...
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Properties of covering spaces replacing base points by contractible subspaces

Let $(X,A)$ and $(Y,B)$ pairs satisfying the homotopy extension property (or a CW-pairs if necessary) with $A$ and $B$ contractible. Let $f:(X,A)\to (Y,B)$ a map of pairs and let $p_X:\widetilde{X}\to ...
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Cell structure of connected sum of complex projective planes

I am trying to find a cell structure for the connected sum $\mathbb{C}P^2\# \mathbb{C}P^2$. My approach is the following: Find a cellular structure for $\mathbb{C}P^2- \text{int}(D^4)$ where $D^4$ ...
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$Y=\prod_{n\in\mathbb{N}}X$.

Let X be a compact CW-complex. The infinite cartesian product $Y=\prod_{n\in\mathbb{N}}X$ is a compact topological space, and as CW-complex It should have a finite number of cells. But in the CW-...
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How can i see a CW-Structure of $S^n$ compatible with the antipodal map?

I dont understand this notion of compatibility. I saw two possibles CW-structures to $S^n$. Please, anyone can explain me this notion?
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Cellular action on CW complex

Let $G$ act cellularly on a CW complex $X$. For each $n\ge 0$, the action induces an action on the indexing set $I_n$ for the $n$-cells. Now look at the cellular chain complex $C_\bullet(X)$. Each ...
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Covering with total space a CW complex

Let $G$ be a discrete group acting properly discontinuous on a CW complex $X$. Does $X/G$ also have a CW structure? Further questions: If not, maybe with stronger conditions ($G$ finite, each ...
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Can a cell-complex have no zero cell?

My question is very simple, but I wasn't able to find an answer in various sources. Cell-complexes are commonly presented using an inductive construction where $n$-cells are attached to $(n-1)$-cells, ...
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Homotopy of continuous map from a space with finite fundamental group

Problem source: 2c on the UMD January, 2018 topology qualifying exam, seen here https://www-math.umd.edu/images/pdfs/quals/Topology/Topology-January-2018.pdf I have an argument for it, but I am not at ...
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Schubert Cell Structure for some variety with prescribed bilinear form

We know the construction of Schubert cell complex structure for the case of Grassmann manifold $Gr_k(\mathbb{C}^n)$ i.e variety of all k- dimensional subspaces $V$ of $\mathbb{C}^n$. Consider that ...
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non-singular variety and CW complex structure

We know non-singular projective variety is a manifold, then it's natural to ask when a non-singular projective(affine) variety can be given a CW-structure? Is there any reference to give a proof?
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When does a map between CW-complexes give a relative CW-complex?

Let $X$, $Y$ be (absolute) CW-complexes and $\varphi:X\to Y$ a continuous map. I would like to know under which assumptions $(Y,\varphi(X))$ is a relative CW-complex. I got interested in this ...
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Homology groups with different complexes

When computing homology groups, I have seen some people use simplicial complexes, delta complexes or CW-complexes. Does it matter which one I use in general? Do they always yield the same homology ...
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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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Name and properties of certain class of subsets of $\Bbb R^n$

I am looking for name or definition of a class of subsets of $ \mathbb{R}^n$ with following properties (in brackets I am describing an alternative set of properties, also interesting for me) : Subset ...
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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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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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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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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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Cellular approximation theorem

It is a well known result that given $n< k $ and a continuous map $f: \mathbb{S}^n \to \mathbb{S}^k$, it can be homotoped to a map $g$ which is not surjective. One way of demostranting that is for ...
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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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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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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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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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The homotopy equivalence of suspension of a CW complex, Hatcher AT exercise $0.25$

Exercise $0.25$ in Allen Hatcher's Algebraic Topology: If $X$ is a CW complex with components $X_\alpha$, show that the suspension $SX$ is homotopy equivalent to $Y\lor_\alpha SX_\alpha$ for some ...
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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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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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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^...