# 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 complexes and manifolds

What is the strongest known theorem (if any) that classify which manifolds can be built as CW complexes? Thank you guys. This is just a question I thought of during class, obviously not homework...
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### Cartesian product of two CW-complexes

Let's $A$ and $B$ are CW-complexes. How to construct CW-complex $A\times B$? Thanks.
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### integral cohomology ring of real projective space

What is the cohomology ring $$H^*(\mathbb{R}P^\infty;\mathbb{Z})?$$ $$H^*(\mathbb{R}P^n;\mathbb{Z})?$$ for mod 2 coefficient, the answer is on Hatcher's book and Proving that the cohomology ring ...
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### Assigning integer to finite CW complex such that following hold. [closed]

For each $n \in \mathbb{Z}$, is there a unique function $\varphi$ assigning an integer to each finite CW complex, such that the following hold? $\varphi(X) = \varphi(Y)$ if $X$ and $Y$ are ...
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### cup product in cohomology ring of a suspension

Let $X$ be a CW-complex. Let $\Sigma$ be suspension. Let $R$ be a commutative ring. Is the cup product of $$H^*(\Sigma X;R)$$ trivial? How to prove? Where can I find the result?
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### Fundamental group of a CW complex only depends on its $2$-skeleton

I was just about to write down my answer to an exercise in algebraic topology and I wanted to use the fact that $\pi_1(X)$ only depends on the $2$-skeleton of $X$ for any CW complex $X$. I am very ...
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### Confusion about topology on CW complex: weak or final?

The topology of the CW complex is defined to be the weak topology: given the sequence of inclusions of the skeleta $X_0 \subseteq X_1 \subseteq_ \cdots$ a subset $A \subseteq X = \cup X_i$ is open iff ...
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### Characteristic map of a n-cell in a CW complex

I have a problem in understanding the purpose of the definition of a CW complex. What really would help me is to understand the following: Let $\sigma$ be a n-cell and $\Phi_\sigma:\mathbb D^n \to X$ ...
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### Visualizing products of $CW$ complexes

I'm learning about products of CW complexes. The sources I've seen talk about the matter as follows: given topological spaces $X$ and $Y$ with a given CW decomposition, we can then form a CW ...
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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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### Cellular Boundary Formula

In Hatcher's book we find, when computing the boundary maps of cellular homology, Cellular Boundary Formula: $d_n(e^{n}_\alpha)=\sum_\beta d_{\alpha\beta}e^{n-1}_{\beta}$ where $d_{\alpha\beta}$ is ...
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### Factorization of a map between CW complexes

I've been working on problem 4.1.16 of Hatcher's Algebraic Topology and am at a complete impasse. The problem is as follows: Show that a map $f:X→Y$ between connected CW complexes factors as a ...
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### Covering of a CW-complex is a CW-complex

Let $X$ be a CW- complex, with filtration $\emptyset \subset X_0 \subset X_1 \subset \cdots \subset X$. Let $p\colon E \to X$ be a covering space. Prove that $E$ is a CW complex with filtration given ...
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### Spaces homotopy equivalent to finite CW complexes

I'm doing a project about Topological Complexity (it doesn't matter what it is for the questions I will ask) and I have proofs for a few results about the bounds of the topological complexity of ...
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### Prove Euler characteristic is a homotopy invariant without using homology theory

I was flipping through May's Concise Course in Algebraic Topology and found the following question on page 82. Think about proving from what we have done so far that $\chi(X)$ depends only on ...
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### The mapping cylinder of CW complex

If $X,Y$ are CW complexes and $f$ a cellular map from $X$ to $Y$, is it true that $M_f$ is a CW complex?
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### Understanding construction of open nbds in CW complexes

I don't understand the construction of open nbds $N_\epsilon(A)$ of $A$ in a CW-complex $X$ given in page 522 of Hatcher's Book. Since the book is available for free online, I'll just copy the entire ...
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### What is the difference between CW-complex and Cellular complex? [closed]

Is every CW-complex is a Cellular space? Is its converse true? If it is true then what is the difference between them? We include the definition of CW-complex in algebraic topology given by Whitehed ...
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### Product of CW complexes question

I am having trouble understanding the product of CW complexes. I know how to actually do the computations and all, I just don't understand how exactly it works. So here's my questions specifically: ...
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### $Mat_{ \infty,n }(\mathbb C)$ 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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### A covering space of CW complex has an induced CW complex structure.

Let $X$ be a $CW$ complex, and let $q : E \rightarrow X$ be a covering map. Prove that $E$ has a $CW$ decomposition for which each cell is mapped homeomorphically by $q$ onto a cell of $X$. Hint: If ...
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### CW complex is contractible if union of contractible subcomplexes with contractible intersection

Exercise 0.23 from Algebraic Topology by Hatcher reads: Show that a CW complex is contractible if it is the union of two contractible subcomplexes whose intersection is also contractible. I have ...
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### Morphism induced by a cellular map between CW-complexes

I'm trying to understand cellular homology as a functor from the category (CW-complexes, cellular maps) to the category of abelian groups sequences. Let $X,Y$ be fixed CW-complexes. My lecturer ...
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### Products of CW-complexes

I am currently reading through May's "Algebraic Topology" and in the chapter on CW-complexes he shows that a product of CW-complexes is again a CW-complex, because one can define product cells using ...
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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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### Try to generalize a problem in Hatcher: finite vs. infinite CW-complexes

While solving a problem in Hatcher I got this doubt in my mind, In the 2nd chapter (Homology) Hatcher asked us to prove the following question... If $X$ is a finite dimensional CW-complex then, a) If ...
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### Quaternion Projective Space $\mathbb HP^n$ and Octonionic Projective Space $\mathbb OP^n$

$\mathbb{R}P^n$ and $\mathbb{C}P^n$ can be built as CW complexes. $\mathbb RP^n = e^0 \cup e^1 \cup \cdots \cup e^n$, $\mathbb CP^n = e^0 \cup e^2 \cup \cdots \cup e^{2n}$. Is there an analogous ...
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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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### A non-positively curved cube complex that admits a local isometric embedding into a Salvetti complex is special.

I am trying to prove the following: "A non-positively curved cube complex $X$ that admits a local isometric embedding (that maps cubes to cubes) into the Salvetti complex of some right-angled Artin ...
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### The motivation of weak topology in the definition of CW complex

Background A CW complex is a Hausdorff space and it is the union of its some of its subsets called cells, and cells are homeomorphic images in $X$ of some closed $k$-balls. The weak topology of a CW ...
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### Homology of orientable surface of genus $g$

I came across the problem of computing the homology groups of the closed orientable surface of genus $g$. Here Homology of surface of genus $g$ I found a solution via cellular homology. This seems ...
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### How does attaching a 1-cell to a path connected CW-complex affect the fundamental group?

If A is the path connected CW complex and X is the new CW complex made by attaching a single 1-cell, is it true that the fundamental group of X is the same as that of A? I've tried to justify this ...
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### If X is a simplicial complex, are the complex of simplicial chains and the one of cellular chains of X identical?

Let $X$ be a simplicial complex. The simplicial chain complex $C_*(X)$ is given by:  C_k(X) := \{\mathbb{Z}-\textrm{combinations of oriented simplices in X, with the identification }\sigma = - \...
Let $X$ and $Y$ be two CW-complexes and $f:X\longrightarrow Y$ be a cellular map. Is $f(X)$ a CW-complex? If the answer is no, then under what conditions $f(X)$ is a CW-complex? Thanks in ...
I have a question about finite cell complexes and compact metrizable spaces. In a paper I read the statement: Let $X$ be a compact metrizable space. Then $X$ is a countable inverse limit \$\varprojlim\...