Questions tagged [at.algebraic-topology]
Homotopy theory, homological algebra, algebraic treatments of manifolds.
1,143
questions
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Is homology finitely generated as an algebra?
If a differential graded algebra is finitely generated as an algebra, is its homology finitely generated as an algebra?
Is it easier if we impose any of the three conditions: characteristic zero; ...
16
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4
answers
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Growth of stable homotopy groups of spheres
Let ${}_2\pi_n^S$ denote the $2$-power torsion subgroup of $n$th stable homotopy group of the sphere spectrum. Its order is a power of $2$: $$|{}_2\pi_n^S|=2^{k_n}.$$
Question: What is known about ...
16
votes
2
answers
766
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Klee's trick --- more applications
In his "Some topological properties..." (1955), Klee gave a construction (simple and beautiful) of an isotopy $h_t\colon\mathbb{R}^{2\cdot n}\to \mathbb{R}^{2\cdot n}$ which moves any compact set $K$ ...
16
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2
answers
859
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Spaces with both "simple homology" and "simple homotopy" at the same time
Maybe every algebraic topology student, at some moment, will ask himself/herself the question: why are $\pi_*$ so difficult and mysterious, especially when compared with (co)homology? Think about the ...
16
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3
answers
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Homotopy Groups of Connected Sums
This was sparked because I wanted to compute $\pi_2(Sym^2(\Sigma_2))$ via $Sym^2(\Sigma_2)\approx \mathbb{T}^4$# $\bar{\mathbb{C}P}^2$.
We know how to compute $\pi_1$ of $M$ # $N$ via van-Kampen's ...
16
votes
1
answer
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Homotopy fiber of a map between classifying spaces
I'm looking for a reference (and precise hypothesis if more are needed) for the following facts (or a correction, if I'm just plain wrong):
Let $G$ and $H$ be topological groups and $f : G \to H$ be ...
16
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10
answers
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Orbifold fundamental group in terms of loops?
In chapter 13 in his notes on 3-manifolds, Thurston defines the orbifold fundamental group to be the group of deck transformations of the universal cover of the orbifold. He also makes a statement "...
16
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8
answers
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Smooth classifying spaces?
Take G to be a group. I care about discrete groups, but the answer in general would be welcome too. There are the various ways to construct the classifying space of G, bar construction, cellular ...
16
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1
answer
910
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Easiest example where pseudo-isotopy fails to be the same as isotopy?
This question concerns diffeomorphism of manifolds. Let $f: M \to M$ be a self-diffeomorphism. We will say that it is isotopic to the identity if there is a continuous one-parameter family of ...
16
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1
answer
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Why does the singular simplicial space geometrically realize to the original space?
I have seen it claimed that (for compactly generated Hausdorff spaces) the geometric realization of the singular (internal) simplicial space is homotopy equivalent to the original space. I know how to ...
16
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2
answers
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Equivariant cohomology vs. invariant cohomology vs. cohomology of quotient space
Given a space $X$ and an action of a group $G$ on $X$, the $G$-invariant cochains with coefficients in an Abelian group $A$ define a sub-cocomplex $\mathcal{C}^{\bullet}_G$ of the cocomplex $\mathcal{...
16
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4
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Coboundaries and Gluing in Cech Cohomology - Intuition?
I'm trying to develop an intuition for Cech cohomology geometrically, but am currently failing. A lot of people seem to say that the groups $H^n$ measure obstructions to gluing local sections to get ...
16
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3
answers
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What is π_1(BG) for an arbitrary topological group $G$?
The classifying space $BG=|Nerve(G)|$ of an arbitrary topological group $G$ does not necessarily have the homotopy type of a CW-complex but the fundamental group should still be accessible. What is $\...
15
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2
answers
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"Economic" CW-structure for Eilenberg-MacLane spaces?
The only really "economic" cell structures for $K(\pi,n)$'s that I know is the one with a single cell in each dimension for $K(\mathbb Z/n\mathbb Z,1)$ and the one with a single cell in each even ...
15
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2
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Every 4-manifold has a $\operatorname{Spin}^c$ Structure
$\DeclareMathOperator\Spin{Spin}\DeclareMathOperator\SO{SO}$I'm having trouble understanding the proof given in Morgan's The Seiberg–Witten Equations and Applications to the Topology of Smooth Four-...
15
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1
answer
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fundamental groups of complements to countable subsets of the plane
This question is a follow-up of this MSE post and a comment by Henno Brandsma:
Question 1. Let $S$ be the set of isomorphism classes of fundamental groups $\pi_1(E^2 - C)$, where $C$ ranges over all ...
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2
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"Strøm-type" model structure on chain complexes?
Background
The Quillen model structure on spaces has weak equivalences given by the weak homotopy equivalences and the fibrations are the Serre fibrations. The cofibrations are characterized by ...
14
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3
answers
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Multiplicativity of Euler characteristic for non-orientable fibrations
Let $E\to B$ be a fibration with fiber F, and assume for simplicity that B is connected. Suppose moreover that B and F have Euler characteristics (perhaps they are manifolds). Then often, one can ...
14
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3
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Infinity-categories vs Kan complexes
It is known (cf. Lurie's book Higher Topos Theory for instance) that higher ($\infty$-) category, in particular topological higher ($\infty$-) groupoids are "better" defined as weak Kan complexes, aka ...
14
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3
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Strictly commutative elements of $E_\infty$-spaces
Let $X$ be an $E_\infty$-space (not necessarily grouplike). Let $x \in \pi_0 X$ be an element; say that $x$ is strictly commutative if there is a map of $E_\infty$-spaces $\mathbb{Z}_{\geq 0} \to X$ ...
14
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Reference Request: Compact manifolds with boundary have the homotopy type of a CW-complex
Let $M$ be a compact manifold (possibly non-smooth) manifold with boundary $\partial M$.
Is the inclusion $\partial M\hookrightarrow M$ homotopy equivalent to the inclusion of a subcomplex into a CW-...
14
votes
1
answer
784
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Is there a category whose isomorphisms are precisely the simple homotopy equivalences?
If we start with the category of finite complexes and continuous maps, and then identify two morphisms iff they are homotopic, we get the homotopy category of finite complexes, and it is trivial to ...
14
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5
answers
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Generalization of winding number to higher dimensions
Is there a natural geometric generalization of the winding number to higher dimensions?
I know it primarily as an important and useful index for closed, plane curves
(e.g., the Jordan Curve Theorem),
...
14
votes
1
answer
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Characteristic classes for odd $K$-theory
There are different models of odd $K$-theory. In one case,
one takes the group $U=\lim\limits_{\longrightarrow}U(n)$ as classifying space. Similarly, if $\mathcal U$ denotes the unitary group of a ...
14
votes
1
answer
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Dijkgraaf-Witten TQFT vs. Representation Theory?
From what I had read, group characters can be "glued" together in a topological fashion and there is something to this effect in the paper by Dijkgraaf and Witten. TQFT seems to be a ...
14
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1
answer
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Are infinite simplicial complexes all manifolds?
Are infinite dimensional simplicial complexes manifolds locally modeled on $\mathbb R^\infty=\operatorname{colim}\mathbb R^n$? If they are homotopy equivalent, are they homeomorphic?
Of course not. ...
14
votes
3
answers
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Strøm model structures on the category of simplicial sets
Let $X,Y$ be simplicial sets. A simplicial homotopy is a simplicial map of the form $h:X\times\Delta^1\rightarrow Y$. There are two distinguished maps
$$
in_0:X\cong X\times\Delta^0\xrightarrow{1\...
13
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3
answers
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Representations of \pi_1, G-bundles, Classifying Spaces
This question is inspired by a statement of Atiyah's in "Geometry and Physics of Knots" on page 24 (chapter 3 - Non-abelian moduli spaces).
Here he says that for a Riemann surface $\Sigma$ the first ...
13
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2
answers
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Well-pointed space which is not locally contractible
I am looking for an example of a well-pointed space in which no (sufficiently small) neighbourhood of the base-point is contractible. As usual, a well-pointed space is a pointed space in which the ...
13
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2
answers
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Is every homology theory given by a spectrum?
Let $E$ be a spectrum. For any CW complex $X$, define $h_*=\pi_i(E\wedge X)$. Then we know that $h_*$ form a homology theory. In other words, there functors satisfy the homotopy invariance, maps a ...
13
votes
1
answer
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Sheaves on Contractible Analytic Spaces
Let $(X,\mathcal{O}_X)$ be a contractible complex analytic space. Suppose that $\mathcal{F}$ is a coherent sheaf of $\mathcal{O}_X$-modules. Can we invoke the fact that $X$ is contractible to conclude,...
12
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4
answers
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The most general context of Mather's Cube Theorems
Quite simply, I'd like to know what is the broadest or most natural context in which either (or both) of Mather's cube theorems hold. If you like, this may mean any of
What properties of $Top$ or $...
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3
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Can triangulations (or some related combinatorial structure) distinguish smooth structures on $RP^4$?
There are exotic versions of $RP^4$, constructed by Cappell-Shaneson, which are homeomorphic but not diffeomorphic to the standard $RP^4$. One way to distinguish them is via the $\eta$ invariant of $...
12
votes
2
answers
645
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Vector bundle for prescribed Stiefel-Whitney classes
I hope this is not trivial.
Let $B$ be a nice topological space (paracompact, CW-complex or whatever you think is nice)
For $i=1,\ldots,n$ let $x_i \in H^i(B,\mathbb{Z}_2)$ be certain cohomology ...
12
votes
1
answer
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What are the applications of Dowker's theorem?
Let $R \subset X \times Y$ be any relation between sets $X$ and $Y$. CH Dowker constructed two simplicial complexes $K$ and $L$ associated to $R$:
a simplex in $K$ consists of finitely many elements $...
12
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2
answers
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Generalized categories for "higher homotopy groupoids"
I was thinking about the definition of higher homotopy groups $\pi_n$ of a topological space in comparison to the common extremely formal fundamental groupoid construction of $\pi_1$. I'd like to be ...
12
votes
1
answer
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Toda's book on homotopy groups of spheres
Yesterday one my friend told about recent book of Hiroshi Toda, where the computations of 3-torsion in homotopy groups of spheres are given up to a very high stem (about 75). This book is published ...
11
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2
answers
880
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When do the Reedy and injective model category structures agree?
Let $R$ be a Reedy category and consider the category $\mathcal{P}(R) = \mathbf{sSet}^{R^{\mathrm{op}}}$ of simplicial presheaves on $R$. When are the Reedy and injective model structures on $\...
11
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2
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Is the geometric realization of a level-wise weak equivalence a weak equivalence?
For the purposes of this question a topological space will mean a compactly generated weak Hausdorff space, though I am actually somewhat flexible on what category of topological spaces we use. I ...
11
votes
1
answer
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Loop spaces motivation
I read that one of the main goals of utilization simplicial methods is to prove that a space is a loop space. On the other hand where lies the main importance to recognize topological spaces as loop ...
11
votes
1
answer
250
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Do spaces admit a weak cogenerating set?
Let $\mathcal C$ be a category. Say that a class of objects $\mathcal S \subseteq \mathcal C$ is weakly cogenerating if the functors $Hom_{\mathcal C}(-,S)$ are jointly conservative, for $S \in \...
10
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3
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Topological dimension versus cohomological dimension
This should be really well known but I don't seem to find a statement about it nor a question in MO answering this.
Consider a Compact Hausdorff topological space $X$. The cohomological dimension of ...
10
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4
answers
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The periodic values in Bott periodicity
After Bott periodicity is proved, one still has to compute the stable values. For the unitary group $U$, this is easy since you can get away with just $\pi_0$ and $\pi_1$. However, I'm having ...
10
votes
1
answer
602
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Tensor products of $\mathbb{E}_\infty$-spaces
In the $\infty$-world, connective spectra play the role of abelian groups, while $\mathbb{E}_\infty$-spaces play that of commutative monoids. This may be rephrased by saying that we may identify the $\...
10
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3
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Spin-H structures
Let us define a Spin-H structure as a reduction of a SO(n)-bundle by the group: $$Spin^H (n)=Spin(n) \times SU(2)/\{ 1,-1\}$$ The Spin-H structures are analogous to the well-known Spin-C structures ...
9
votes
2
answers
699
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Correspondence between operads and monads requires tensor distribute over coproduct?
In checking the details of the correspondence between operads over a symmetric monoidal category and monads on some associated endofunctor of the category, I cannot make the obvious proof work without ...
9
votes
2
answers
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"Skew Cohomology" of a Space
Let $X$ be a space. The symmetric group $\Sigma_{n+1}$ acts on the function space
$$
X^{\Delta^n}
$$
of continuous maps from the standard $n$-simplex to $X$. The action is induced
by permuting the ...
9
votes
0
answers
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Explicit description of a subgroup of the braid group $\mathsf{B}_2(C_2)$
This is related to my previous MathOverflow question Fundamental group of $\mathrm{Sym}^2(C_g)$ minus the diagonal.
Let $C_2$ be a smooth curve of genus $2$ and $X:=\mathrm{Sym}^2(C_2)$ its second ...
9
votes
3
answers
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Does a connected manifold with vanishing Euler characteristic admit a nowhere-vanishing vector field?
A version of the "hairy ball" theorem, due probably to Chern, says that the Euler-characteristic of a closed (i.e. compact without boundary) manifold $M$ can be computed as follows. Choose any vector ...
9
votes
1
answer
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The connective $k$-theory cohomology of Eilenberg-MacLane spectra
Consider the connective $K$-theory spectrum $ku$. Let $H\mathbb{Z}$ be the Eilenberg-MacLane spectrum and $H\mathbb{F}_p$ be the mod-$p$ Eilenberg-MacLane spectrum.
Is it known what $ku^{*}(H\...