Questions tagged [cw-complexes]
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167
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"Almost embedding" the complete 2-dimensional complex $\mathcal K_7^2$ into $\Bbb R^4$
Let $\mathcal K_7^2$ be the complete 2-dimensional simplicial complex on seven vertices, i.e. it has all $7\choose 2$ edges and all $7\choose 3$ 2-simplices (and no higher-dimensional simplices).
I ...
- 12.8k
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What is like "flat" but for based connected CW-complexes?
Let $X$ be a based connected CW-complex $X$.
Say that $X$ is CW-flat if, for each map of based connected CW-complexes $f \colon Y \to X$ such that $\pi_n(f)$ is injective, $\pi_n X \wedge f$ is also ...
- 21
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Relationship between quotient CW-complexes after attaching cells
I have been trying to prove the following simple-looking result which I require for some work in low-dimensional topology. I expect it is likely true and in a textbook somewhere so any reference or ...
- 187
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"Star" of a CW-complex
Suppose we have a CW-complex $X$ with a 0-cell $e^0$. Is the union of all the cells (of higher dimensions) for which $e^0$ is a boundary point open in $X$?
I don't know if it has a name, but a similar ...
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Is the Whitehead bracket $\pi_{p}(X)\otimes \pi_{q}(Y)\to \pi_{p+q-1}(X\vee Y)$ injective?
Let $X$ and $Y$ be finite CW-complexes and $p,q\geq 2$. The Whitehead bracket induces a homomorphism $\pi_{p}(X)\otimes \pi_{q}(Y)\to \pi_{p+q-1}(X\vee Y)$, $\alpha\otimes \beta\mapsto [\alpha,\beta]$....
- 497
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A cell complex constructed from singular knots
Let $\mathcal K_n$ be the set of all $n$-singular knots up to isotopy,i.e. an immersion of $S^1$ into $\mathbb R^3$ with $n$ transverse double points that is an embedding when restricted to the ...
- 141
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Replacing a $G$-CW-complex with a $G$-homotopy equivalent $G$-simplicial complex - can anyone supply a reference?
Let $G$ be a group (not a topological group, just a group). By a $G$-complex I mean a CW-complex with an action of $G$ that takes cells to cells so that the pointwise and setwise stabilizer of each ...
- 1,588
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Minimal cell structures in combinatorial model categories
I recently rediscovered a classical theorem from Hatcher which states that simply-connected CW complexes have a 'minimal' cell structure, where the cells correspond to spheres and disks indexed by the ...
- 285
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Homotopy groups of finite CW complex finitely generated as Lie algebra
This is probably a well-known question, but I haven't found the answer on MO or MSE.
It is well-known that the homotopy groups of a finite CW complex $X$ need not be finitely presented, even as $\...
- 27.9k
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Explicit CW-complex replacement of the space of reparametrization maps
Let $P$ be the space of nondecreasing surjective maps from $[0,1]$ to itself equipped with the compact-open topology: $P$ is contractible. There exists a trivial fibration $P^{cof} \to P$ from a CW-...
- 3,821
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Differential graded modules and the Serre-Swan theorem
I am thinking about how connections combine with a modification of the Serre-Swan theorem, which relates vector bundles to projective modules.
If $E \rightarrow B$ is a vector bundle, or even just any ...
user30211
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173
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The monoid of stably-free modules over integral group rings
Fix a torsion-free group G, let $M_G$ be the monoid of stably-free $\mathbb{Z}G$-modules under operation $\oplus$, the direct sum of modules.
In studying objects related to Wall’s D2 problem on CW-...
- 187
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The number of $n$-cells attaching to $K^{n-1}$ in Wall's construction
Let $\phi:K\to X$ be a map, with mapping cyliner $M=X\cup_{\phi}(K\times I)$. We define $\pi_n (f)$ as $\pi_n (M,K\times 1)$. An element of $\pi_n (f)$ is represented by a pair of maps $\beta :S^{n-1}\...
- 277
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Spectral sequence in Adams's book, Theorem 8.2
I am having trouble in understanding Theorem 8.2 of Adams's book and the application afterwards of constructing the spectral sequence. I think I should prove somehow that the spectral sequence in this ...
- 224
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390
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Homotopy groups of mapping cylinder
Let $f:K\to X$ be a map, with mapping cyliner $M=X\cup_{f}(K\times \{ 1\})$. We define $\pi_n (f)$ as $\pi_n (M,K\times \{ 1\})$. An element of $\pi_n (f)$ is represented by a pair of maps $\alpha :S^{...
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