Questions tagged [at.algebraic-topology]
Homotopy theory, homological algebra, algebraic treatments of manifolds.
8,257
questions
4
votes
1
answer
324
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Simplicial model categories and simplicial equivalence
Let $M$ and $N$ two very nice simplicial model categories and let $F:N\rightarrow M$ be a (nice) simplicial functor which induces an equivalence of homotopy categories, i.e. $Ho(F): Ho(N)\rightarrow ...
- 511
5
votes
2
answers
416
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On spaces with finite homological dimension
Let $X$ be a connected $CW$-complex, such $\pi_1(X)$ is torsion-free and $H_k(X,\mathbb Z) = 0$ for all $k \geq N$ and some $N \in \mathbb N$. Then
$(1)$ Does it follow that $X$ is homotopy-...
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3
votes
1
answer
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How to compute $\pi_0$ of $Maps(S^1, \Omega^2({S}^2, p))$
Denote by $\Omega^2({S}^2)$ the space of DOTTED maps from the $2-$sphere $S^2$ onto itself. And consider its FREE loop space $X=\mathcal{L}(\Omega^2({S}^2))=Maps(S^1, \Omega^2({S}^2))$. I think that $\...
- 10.7k
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votes
3
answers
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Find a surface or 3-manifold whose fundamental group is $(\mathbb{Z}/n\mathbb{Z}) \rtimes (\mathbb{Z}/2\mathbb{Z})$
I know by Van Kampen's Theorem that we can obtain $\pi_1(S_1 \vee S_1) = \mathbb{Z} * \mathbb{Z}$, so I am wondering if we can construct a surface or 3-manifold whose fundamental group is $\mathbb{Z}...
- 5,231
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votes
0
answers
163
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Seminaire Henri Cartan: Eilenberg-Maclane spaces
Consider Theorem 1 on page 11-09 of Seminaire Henri Cartan 7e annee: 1954/1955. The theorem is about a homomorphism $f:H(X)\rightarrow H_*(K(\pi,n);\mathbb{Z})$, where $X$ is a differential graded ...
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votes
0
answers
325
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Fundamental groupoid and fibration
In this post, it is said that a functor from the fundamental groupoid of a space $X$ (denoted by $\Pi(X)$) to the category $\mathrm{Vect}$ of vector spaces gives a flat vector bundle over $X$. But I ...
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1
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113
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a compact set with nonempty convex sections
Let $X = [0,1]^d$ be the unit cube in the $d$-dimensional Euclidean space.
For every $x \in X$ and every coordinate $i=1,2,\ldots,d$ denote by $x_{-i} := (x_j)_{j \neq i}$.
Given a set $Y \subseteq X$ ...
- 627
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votes
1
answer
908
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The homology groups of the smooth locus of a singular variety
Let $X$ be a complex irreducible variety and denote its smooth locus by $X^{smooth}$. I would like to know what can be said about the induced maps $H_i(X^{smooth};\mathbb{Q})\rightarrow H_i(X;\mathbb{...
- 4,091
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votes
2
answers
685
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Connectivity of suspension-loop adjunction
Let $X$ be a $k$-connected spectrum for $k \in \Bbb{Z}$.
I want to deduce how connected the counit of $(\Sigma^\infty, \Omega^\infty)$- adjunction is, that is, how connected is the map
$$
\Sigma^\...
- 18.6k
4
votes
1
answer
256
views
A question about Wall's construction for CW-complexes
For a given map $\phi :X\longrightarrow Y$, the mapping cylinder of $\phi$ is defined by $M_{\phi}:=Y\bigcup_{\phi} (X \times \{ 1\})$. Denote $\pi_n (M_{\phi},X \times \{ 1\} )$ by $\pi_n (\...
- 3,431
2
votes
1
answer
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views
How does the high-dimensional combinatorial Laplacian work?
When considering the boundary and coboundary maps, we have the common definitions that the boundary map based on the space of chains $C_k(X)$ is $$\partial_k([v_0,...,v_k])=\sum_{i=0}^k (-1)^i[v_0,...,...
- 3,367
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votes
0
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Quotient space, a fundamental group, and higher homotopy groups 2
Previously, I ask for comments/suggestions on setting up the calculation in Quotient space, homogeneous space, and higher homotopy groups. There, however, I was looking for whatever methods and tools ...
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3
votes
0
answers
857
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Quotient space, homogeneous space, and higher homotopy groups
Preparation and my input:
For the quotient space $G/H$, knowing the homotopy
groups of $G$ and $H$ one can determine homotopy groups from the long
exact sequence
$$
...
\to \pi_n(H) \to \pi_n(G) ...
- 10.3k
4
votes
1
answer
151
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The homological negligibility of certain subsets in compact manifolds
Let $n\ge 3$ and $X$ be a compact connected $n$-manifold (without boundary).
I need a reference to the following facts (which I believe are true at least in dimension $n=3$):
Fact 1. For every ...
- 17.2k
11
votes
3
answers
809
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Induced maps on homotopy groups by self maps of $\mathbb{CP}^n$
Let $f:\mathbb{S}^2\to \mathbb{S}^2$ with degree $d$.
It is well known that the induced map $$f_\ast:\pi_3(\mathbb{S}^2)=\mathbb{Z}\to \pi_3(\mathbb{S}^2)=\mathbb{Z}$$ is given by multiplication by $...
- 5,016
24
votes
1
answer
558
views
Action of the degree 2 map on $\pi_8(S^4)$
I am currently reading Sullivan's Geometric Topology: Localization, Periodicity, and Galois Symmetry, on page 34 Sullivan claims that the degree 2 map $2:S^4 \to S^4$ induces the map $\left(\begin{...
- 5,016
3
votes
1
answer
255
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Poincaré dual of the generators of $H^d(\mathbb{RP}^5,\mathbb{Z}_2)$
We know $H^d(\mathbb{RP}^5,\mathbb{Z}_2)=\mathbb{Z}_2$. So there are two classes of $\mathbb{Z}_2$ generators, trivial and nontrivial, for $d=0,1,2,3,4,5$.
Wha are the Poincaré dual $(5-d)$-...
- 10.3k
4
votes
0
answers
66
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Irreducible separators of compact manifolds
Definition. A closed subset $S$ of a topological space $X$ is called
$\bullet$ a separator of $X$ if $X\setminus S$ is disconnected;
$\bullet$ an irreducible separator if $S$ is a separator of $X$ ...
- 40.9k
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votes
1
answer
488
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can $H^*(\mathbb{C}P^n;\mathbb{Z})$ be the cohomology of some Eilenberg-Maclane space $K(\pi,1)$?
Recently I came across the following question:
can $H^*(\mathbb{C}P^n;\mathbb{Z})$ be the integral cohomology ring of some Eilenberg-Maclane space $K(\pi,1)$?
I guess (without strong evidences) that ...
- 13.5k
14
votes
1
answer
1k
views
CW complex of iterated loop spaces
In Milnor's book Morse Theory, it is proved that the loop space $\Omega S^n$ of the n sphere has the homotopy type of a CW complex with one cell each in the dimensions 0, n-1, 2n-2, 3n-3, ... Or more ...
- 10.7k
4
votes
0
answers
237
views
Non-spin 5-manifold and $2^2$-Bockstein homomorphism
The $2^2$-Bockstein is $\beta_4$ is associated to
$$0\to\mathbb{Z}/2\to\mathbb{Z}/{8}\to\mathbb{Z}/{4}\to 0,$$
(The $2^n$-Bockstein homomorphism
$$\beta_{2^n}:H^*(-,\mathbb{Z}/{2^n})\to H^{*+1}(-,\...
- 10.3k
7
votes
2
answers
829
views
Moore decomposition, dual to Postnikov tower
Let $X$ be a CW complex with given cohomologies $H^n(X; \mathbb{Z}) = G$, $H^m(X; \mathbb{Z}) = H$ and other reduced cohomologies are zero. Which additional algebraic information/structures do I need ...
- 18.8k
0
votes
1
answer
179
views
detecting weak equivalences in a simplicial model category
Suppose that we have a simplicial model category $M$. The simplicial enrichment will be denoted by $map_{M}$. Let $f:A\rightarrow B$ be a morphism in the category $M$ such that $A$ is cofibrant. ...
- 29.8k
4
votes
0
answers
195
views
What are the compact objects in the homotopy category of diagram spectra?
Let $\mathcal{C}$ be a discrete index category and $\mathcal{S}$ the category of (e.g.) orthogonal spectra. We consider the homotopy category $\mathrm{Ho}(\mathrm{Fun}(\mathcal{C,S}))$ of diagram ...
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101
votes
3
answers
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views
What is the mistake in the proof of the Homotopy hypothesis by Kapranov and Voevodsky?
In 1991, Kapranov and Voevodsky published a proof of a now famously false result, roughly saying that the homotopy category of spaces is equivalent to the homotopy category of strict infinity ...
- 40.2k
5
votes
1
answer
647
views
Localization of a model category
Let $M$ be a very nice model category (cofibrantly generated, combinatorial or cellular and left proper simplicial model category). Let $f: X\rightarrow Y$ and $g: X\rightarrow Z $ be two morphisms ...
- 4,410
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votes
4
answers
1k
views
Complements of Simply Connected Subsets of the Plane
this is my first question here! Hopefully it is appropriate. Let $\mathbb{A}$ be the punctured plane, i.e. the 'standard' annulus. For compact, connected subsets of the plane (planar continua) $X \...
- 131
9
votes
1
answer
686
views
Del Pezzo surfaces and Picard-Lefschetz theory
Let $X$ be a smooth compact del Pezzo surface. For instance, one can consider the most classical case of a cubic surface. It is well known that the Picard lattice of $X$ is related to a root system (...
- 6,945
26
votes
1
answer
3k
views
connectivity of the group of orientation-preserving homeomorphisms of the sphere
In the paper "Local Contractions and a Theorem of Poincare" Sternberg has mentioned the following question which was open when the paper was written:
Is the group of orientation-preserving ...
- 9,349
30
votes
1
answer
757
views
Is a filtered colimit of rational spaces again rational?
Let me first explain the statement of the question and then give some indication why the answer might be 'yes'.
By a space I mean, say, a simplicial set and by rational I mean rational in the sense of ...
- 51.5k
8
votes
1
answer
330
views
Lifting Strict Comonoids and Comodules to Quasicategories
$\newcommand{\M}{\mathcal{M}}$
Suppose I have a monoidal simplicial model category in which every object is cofibrant $(\M,\otimes,\mathbb{1})$ and I want to look at its underlying monoidal ...
- 10.1k
25
votes
2
answers
2k
views
Did Peter May's "The homotopical foundations of algebraic topology" ever appear?
In the monograph Equivariant Stable Homotopy Theory, Lewis, May, and Steinberger cite a monograph "The homotopical foundations of algebraic topology" by Peter May, as "in preparation." It's their [107]...
- 30.2k
3
votes
2
answers
326
views
Good, detailed references for "mod p lower central series"
I am looking for good, detailed references for "mod $p$ lower central series".
So far I only find papers such as (https://core.ac.uk/download/pdf/81193793.pdf, https://www.sciencedirect.com/science/...
CommunityBot
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votes
1
answer
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Component and quasi-component
Let $X$ be a topological space and $x\in X$. Then the quasi-component of the point $x$, denoted by $C_x$, is the intersection of all clopen (closed-and-open) subsets of $X$ which contain the point $x$...
- 1,269
2
votes
0
answers
74
views
Does there exist a "Margolis-type" definition of equivariant cellular towers?
I am interested in cellular towers in the equivariant stable homotopy category $SH_G$ corresponding to a compact Lie group along with a complete universe for it.
Note here that a cellular tower for ...
- 16.5k
20
votes
1
answer
883
views
Double Counting: Motivic Edition
One of the most important proof techniques in combinatorics is double counting: proving that both sides of an identity count elements of some set in two different ways. This question is an attempt at ...
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10
votes
1
answer
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Etymology of 'spectrum' in algebraic geometry and algebraic topology
In algebraic geometry, one has the notion of the spectrum of a commutative ring. These spectra serve as local charts for schemes.
In algebraic topology, a spectrum is a sequence of pointed spaces $...
- 39.3k
4
votes
1
answer
219
views
How are p-primary parts determined for odd p?
When looking at surveys of computations of the homotopy groups of spheres there is a common theme. All the odd primary parts are thrown away.
How are odd primary part calculations done in relation ...
- 51.5k
2
votes
1
answer
243
views
Do Mackey (co)homology functors factor through derived categories? References with details?
Let $G$ be a compact Lie group; I will write $SH_G$ for the (equivariant) stable homotopy category of $G$-spectra (say, with respect to a complete universe; does its choice affect the homotopy ...
- 55k
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votes
0
answers
393
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A cell decomposition of a CW-complex and, stratification of a topological space
What is the difference between the notion of cell decomposition of a CW-complex, and the notion of stratification of a topological space ?
I know that cell decomposition of a CW-complex is usefull to ...
- 325
3
votes
1
answer
342
views
If $X, Y$ are topological spaces, with $Y$ being a k-space, and $f : X \to Y$ is a proper covering map, is $X$ necessarily a k-space?
A k-space is a compactly generated Hausdorff topological space. (I used the terminology "k-space" in the question, in order keep the question within the limit of 150 characters.)
Note that under the ...
- 11.4k
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votes
0
answers
1k
views
Original references for the homotopy groups $\pi_5(SU(3))$ and $\pi_4(SU(2))$?
For revision of a paper (http://arxiv.org/abs/1008.1189), I'd like to correct my references to the original work on aspects of the homotopy groups $\pi_5(SU(3))$ and $\pi_4(SU(2))$. I'm not a ...
- 18.8k
3
votes
0
answers
82
views
Reference Request: Equivariant Symplectic bordism
Non-equivariantly, symplectic bordism has been developed extensively by Ray, Gorbunov, and specially S. Kochman in this memoir: http://dx.doi.org/10.1090/memo/0496 Yet the coefficients ...
- 6,766
21
votes
5
answers
5k
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Is there a good way to understand the free loop space of a sphere?
I'd like to understand the structure of the free loop space of $S^n$ for small values of $n$. Here "understand" means roughly that I'd like to know a CW complex with the same homotopy type.
I ...
- 7,373
16
votes
0
answers
734
views
What would be the simplest analog of Langlands in algebraic topology?
It is oversimplified, I know, but just as a superficial analogy, one may think of the fact that abelianization of the fundamental group is the first homology group, as some remote relative of class ...
- 17.5k
3
votes
1
answer
429
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non-existence of global coordinates
Assume we have a smooth manifold, $M$, of dimension $n$. (An example of interest is the case when $M$ is a compact and orientable Riemann surface of genus $g$, but the question is intended to be broad....
- 25.6k
6
votes
0
answers
126
views
Localizations of group algebras of free groups
$\newcommand{\QQ}{\Bbb Q}$
Let $G$ be a free group on the symbols $x_1, \dots, x_n$, with $\QQ[G]$ its rational group algebra.
Write $\varepsilon: \QQ[G] \to \QQ$ for the augmentation, and for $\...
- 51.5k
6
votes
0
answers
224
views
The chromatic splitting conjecture and functoriality
Let $M$ be a finite spectrum, so that $L_{K(n)} M = M \wedge L_{K(n)} S$. Recall that (a weak version of) the chromatic splitting conjecture states that the chromatic attaching map $L_{n-1} M \to L_{n-...
- 5,550
18
votes
2
answers
582
views
primary decomposition for nonabelian cohomology of finite groups
Let $G$ be a finite group, and let $M$ be a group on which $G$ acts (via a homomorphism $G\to \operatorname{Aut}(M)$).
If $M$ is abelian, hence a $\mathbb{Z}G$-module, there is a primary ...
- 35k
7
votes
2
answers
726
views
What does it mean to say the first Goodwillie derivative of $TC$ is $THH$?
A paradox:
Goodwillie calculus considers only finitary functors.
$TC$ isn't finitary.
Yet in some sense $\partial(TC) = \partial(K) = THH$ is the crux of the Dundas-Goodwillie-McCarthy theorem.
(...
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