22
votes
Does there exist a complete algebraic invariant of the homotopy type of a finite CW-complex?
Following the Peter's suggestion, I'll turn my comment in an answer.
In general by algebraic category one means variety in the sense of universal algebra, or category of models of a (possibly ...
- 9,111
19
votes
Is any CW complex with only finitely many nonzero homology groups homotopic to a finite dimensional CW complex?
In the simply connected case, the answer is yes.
In the general case, the theory was worked out in complete detail by Wall in the paper:
Wall, C. T. C. Finiteness conditions for CW-complexes. Ann. of ...
- 18.8k
19
votes
Acyclic group and finite CW-complex
I presume by "acyclic" you are referring to homology with $\mathbb{Z}$ coefficients. There are many such examples.
For instance, you can take two elements $u,v$ in the free group $F_2$ of rank 2 ...
- 25k
18
votes
Accepted
Acyclic group and finite CW-complex
The Higman group with presentation
$$\langle{a,b,c,d}\mid{aba^{-1}b^{-2}},~bcb^{-1}c^{-2},~cdc^{-1}d^{-2},~
dad^{-1}a^{-2}\rangle$$
is perfect, and the 2-complex associated to this presentation
has ...
- 496
17
votes
Accepted
Is the decomposition of the homotopy type of a complex into a bouquet unique?
In Hilton&Roitberg paper "On principal $S^3$-bundles over spheres" it's proven that if you have a prime order $p \neq 2,3$ class $\alpha$ in $\pi_k(S^n)$ that is a suspension, then for a ...
- 4,600
16
votes
Finite CW complex with finite non-abelian fundamental group and higher homologies zero
Take any group $G$ (non-abelian if you like) that has a presentation $G = \langle g_1, \dots, g_s \ | \ r_1, \dots, r_{s} \rangle$ with the same number of generators and relations.
Form a CW-complex $...
- 11.9k
15
votes
Accepted
Are Spanier-Whitehead duals of general spaces expressible through some generalization of normal bundles?
Let $X$ be a finite complex. Then the functor
$$\lim_X:\operatorname{Fun}(X,\operatorname{Sp})\to \operatorname{Sp}$$
sending a local system of spectra $E$ to its limit preserves all colimits. Indeed ...
- 16.5k
15
votes
Accepted
Can we embed a closed manifold into a homotopy equivalent CW complex?
Pick a torus, and add two discs along a meridian and a longitude. You get a 2-complex homotopic to a sphere that does not contain a sphere. This generalises easily to any genus by picking a genus-$g$ ...
- 10.5k
14
votes
Accepted
loop space of a finite CW-complex
This is true for finite $\pi_1$ and false for infinite $\pi_1$: Let $\widetilde{X}$ denote the universal cover of $X$, then $\Omega\widetilde{X}$ is the unit connected component of $\Omega X$, and $\...
- 10.8k
14
votes
Accepted
Delta-generated spaces vs CW complexes
The category of $\Delta$-generated spaces is quite large as it includes all first countable, locally path connected spaces. Hence, all compact, connected, locally connected subsets of $\mathbb{R}^n$ ...
- 7,193
13
votes
Accepted
How can I endow a "locally product" CW structure on a vector bundle over a CW complex?
The authors of this book are attempting to use CW structures to justify certain cohomology isomorphisms, but this seems to be the wrong approach since some of their claims about CW structures are just ...
- 20k
13
votes
Accepted
Homotopy groups of finite CW complex finitely generated as Lie algebra
I think Ian Leary's answer to this question gives a counterexample. His construction shows that for every $k\ge 2$ there exists a group $G_k$, and a finite $k$-dimensional CW complex $X$ such that $\...
- 10.9k
12
votes
Is any CW complex with only finitely many nonzero homology groups homotopic to a finite dimensional CW complex?
As requested I am writing this as an answer.
No there are spaces with vanished homology which are not homotopy equivalent to finite CW-complexes.
For example if $G$ is an acyclic group, then the ...
- 27.5k
12
votes
Accepted
Weak homotopy equivalence between $\Omega \underset{\rightarrow}{\lim}Z_n$ and $\underset{\rightarrow}{\lim}\Omega Z_n$
First note that Hatcher's exercise says "where direct limits mean mapping telescopes", so he is defining $\underset{\rightarrow}{\lim}$ to mean the telescope. I disapprove of that quite strongly. ...
- 56.9k
10
votes
Allowing $G$-CW complexes to have more general cells
Let me start with the fact that, in one sense, it's true that Type 1 complexes are all that are "needed." That's true in the sense that complexes built from Type 2 and 3 cells have the $G$-homotopy ...
- 2,062
10
votes
Accepted
$G$ uncountable implies $K(G,1)$ is not a finite CW complex
For any finite CW-complex $X$ and any basepoint $x \in X$, the fundamental group $\pi_1(X,x)$ is finitely presented. (This is a consequence of the Seifert-van Kampen theorem.) In particular, the group ...
- 52.6k
10
votes
Accepted
Is every open topological $d$-manifold homotopy equivalent to a CW-complex of dimension $\leq d-1$?
$\DeclareMathOperator{\co}{H}
\DeclareMathOperator{\ch}{C}
\newcommand{\zz}{\mathbb{Z}}
\newcommand{\nn}{\mathbb{N}}
\newcommand{\A}{\mathcal{A}}
\newcommand{\B}{\mathcal{B}}
\DeclareMathOperator{\lf}{...
- 1,726
10
votes
Accepted
Whitehead product and a homotopy group of a wedge sum
Here are some details which are related to Tyler's comment.
I recommend looking at the paper "Induced Fibrations and Cofibrations" by Tudor Ganea (1967). For connected based spaces $X$ and $...
- 18.8k
10
votes
Is the decomposition of the homotopy type of a complex into a product and into a smash product unique?
Let me write $S^n/p$ for the cofibre of $p$ times the identity map on $S^n$, or in other words the mod $p$ Moore space with homology in degree $n$.
If $p$ and $q$ are coprime then $S^n/p\wedge S^m/q$ ...
- 56.9k
10
votes
Accepted
Contractible subcomplex containing 1-skeleton?
I think the answer is no.
In Hatcher's Algebraic Topology there is an example of an acyclic 2-dimensional complex with one 0-cell, two 1-cells and two 2-cells (Example 2.38). You start with a wedge of ...
- 10.9k
10
votes
Connectivity of fibers under fibration replacement
No, this isn't true.
Fix $k \geq 1$ and let $Y$ be the join $S^k \ast [0,1]$, which is homeomorphic to $D^{k+2}$. It is a simply-connected CW-complex and has trivial homotopy groups.
Let $X$ be the ...
- 52.6k
9
votes
Generalized cohomology of CW complex is direct limit?
In general, no. Assuming, say, that the structure maps $SE_n\to E_{n+1}$ are inclusions, the correct statement is
$$
E^n(X) \cong [X,\lim_k \Omega^k E_{n+k}],
$$
and if $X$ is not compact that is not ...
- 2,062
9
votes
Accepted
How can I construct a closed manifold from a finite CW complex?
Take $X=S^3$. Then no closed manifold of dimension at least 6 has the same homotopy type.
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8
votes
Accepted
Covering with Deck group $\mathfrak{S}_3$
Here is a picture from Topology and Groupoids
It is meant to show in (i) the Cayley graph of the presentation $\mathcal P$ of $G=S3$, $\{x,y:x^3,y^2,xyxy\}$. The Cayley graph is the $1$-skeleton ...
- 12.3k
8
votes
How can I construct a closed manifold from a finite CW complex?
More generally, suppose $n \le m$ are non-negative integers,
$X$ is a CW complex of dimension $\le n$, $M$ is a non-empty, closed $m$-manifold,
and $X$ and $M$ have the same homotopy type.
It is well ...
- 18.8k
8
votes
Accepted
Spectral sequence in Adams's book, Theorem 8.2
This is not a complete answer, but it is too long for a comment. I will assume you are interested in the spectral sequences in Section 8 of Adams, not the Atiyah-Hirzebruch spectral sequences of ...
- 4,254
7
votes
Accepted
CW Product via Whitehead map
The attaching map of the product of cells is sometimes described as an exterior join construction.
Let $F:D^n\to Cf\subseteq X$ be the characteristic map of an $n$-cell of $X$ with attaching map $f:S^...
- 35.9k
7
votes
Accepted
Turning injection of homotopy groups to an isomorphism
Your question is equivalent to the following:
Given a cellular inclusion $i : X\to Y$, when is there a retraction $r:Y \to X$?
(Being a retraction means that $r\circ i: X\to X$ is the identity.)
...
- 18.8k
7
votes
Accepted
Attribution of theorem saying that inducing isomorphism on homology implies homotopy equivalence between H spaces that are CW complexes
The result for simply-connected spaces is usually attributed to J.H.C. Whitehead, in particular Theorem 14 in
J.H.C Whitehead. Combinatorial homotopy II. Bull. Amer. Math. Soc. 55, (1949). 453–496.
...
- 17.4k
7
votes
Is any CW complex with only finitely many nonzero homology groups homotopic to a finite dimensional CW complex?
Supplementing Chris's answer, you could take $G$ to be any acyclic group of infinite cohomological dimension. Such groups exist, e.g. binate groups. See
Berrick, A.J., The acyclic group dichotomy., J....
- 35.9k
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