Questions tagged [at.algebraic-topology]
Homotopy theory, homological algebra, algebraic treatments of manifolds.
8,228
questions
6
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0
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206
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Can representable presheaves be made injectively fibrant?
I suspect that the answer to my question is no, but let me give it a shot anyway.
If $\mathcal{A}$ is a small simplicially enriched category, then the category of simplicial presheaves $\mathsf{sSet}^...
0
votes
0
answers
101
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group actions preserve the cup product
Let $X$ be an oriented compact manifold of dimension $2k$. Suppose that a compact Lie group $G$ acts differentiably on $X$ in an orientation-preserving way. Let $B$ be the ${\mathbb R}$-bilinear form ...
15
votes
1
answer
376
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Relations among Pontryagin numbers
The Hattori-Stong theorem describes the image of the morphism:
$$\tau:\Omega^{SO}_*\rightarrow H_*(BSO;\mathbb{Q})$$
that associates to any closed, smooth, oriented manifold $M$ the homology class $\...
6
votes
1
answer
619
views
Cohomology of function spaces
Let $M,N$ be smooth manifolds and $C^\infty(M,N)$ be the function space with Whitney topology.
If we know the cohomology groups of $M,N$, can we calculate the cohomology groups of $C^\infty(M,N)$?
9
votes
2
answers
704
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Four manifold without point homotopy equivalent to wedge of two-spheres?
Let $X$ be a closed, simply-connected four-manifold. Let $X'$ be obtained from $X$ by removing a point. Is $X'$ homotopy equivalent to a wedge of $S^2$s?
3
votes
1
answer
271
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K-groups of strict henselization of stalks
How well are the algebraic K-groups of the strict henselization of the stalks $\mathcal{O}_{X,p}^{sh}$ at geometric points of a scheme $X$ understood? I am particularly interested in the case of ...
3
votes
0
answers
301
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The Baum Connes Conjecture - the approach using spectra
In this article James Davis and Wolfgang Lück introduce a approach using spectra to formulate the Baum Connes Conjecture for a discrete group $G$. In order to do so, they construct a functor
$$\...
4
votes
1
answer
367
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Torsion In $K$ theory on simply connected manifolds
The usual construction for finding torsion elements on complex $K$ theory is using flat vector bundles. So is it still possible to find a simply connected compact space with a nonzero torsion in its $...
1
vote
1
answer
167
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Colocal Objects in Enriched Bousfield Colocalizations
Let $C$ be a $V$-model category, and $\mathcal{K}$ a set of objects of $C$.
Let me denote (derived) simplicial homotopy function complexes by $\text{Dmap}$
and derived $V$-function complexes by $\text{...
4
votes
0
answers
173
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Homotopy invariance of the moduli stack of smooth $G$-bundles?
This question ought to have a straightforward (perhaps even glaringly obvious) answer, but so far I've already wasted a few hours trying to untangle this web of inconsistent identifications. I'm sure ...
4
votes
1
answer
188
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Decomposition of $\Gamma$-modules into simple objects
Let $\Gamma$ be the category of finite pointed sets. The abelian category $\mathrm{Mod-}\Gamma$ is the category of functors $\Gamma^{\mathrm{op}} \to \mathrm{Vect}_k$, where $k$ is a field (see ...
10
votes
1
answer
540
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An equivariant social choice in Mathematical economics
Motivated by this paper and its economics motivations, we recall that a social choice among $n$ objects is a continuous function $$f:\overbrace{M\times M\times\cdots\times M}^{\text{$n$ times}}\to ...
8
votes
1
answer
454
views
Cartan formula for Steenrod squares on the cochain level
Steenrod originally defined his squares using explicit cochain-level formulas for simplicial mod-2 cochains. To this end, he introduced higher cup products, which control the failure of the usual cup ...
16
votes
2
answers
2k
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Is the Gromov conjecture still open?
Today I read about Gromov's definition of minimal volume for smooth manifolds.
$$\min {\rm Vol}(M):=\inf_{|K_g|\leq1}\{{\rm Vol}(M,g)\}.$$
Gromov's conjecture states that for every closed simply ...
4
votes
0
answers
127
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Spin bordism with non free involution
Is there a comprehensive account of GEOMETRIC equivariant spin bordism groups with respect to the group $ \mathbb{Z}/2$ (instead of homotopy theoretical trough equivariant Thom Spectra),...
13
votes
0
answers
364
views
What is the cup-product structure like on a hyperbolic 5-manifold?
Let $X$ be a hyperbolic 5-manifold. Can there be any class in $H^2(X;\mathbf{Z})$ that is torsion and whose square in $H^4(X;\mathbf{Z})$ is not zero?
For example, are there hyperbolic 5-manifolds ...
6
votes
0
answers
232
views
Isomorphic copies of the real line--can these isomorphisms be made explicit?
This question has bugged me for a long time. I've asked some of my professors and they seem to believe that these objects haven't been studied. I'm prone to believe that the construction is too simple ...
6
votes
1
answer
446
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Restrictions on $\pi_1(X)$ of geometric origin (Kähler groups as example)
There's and old and extensively studied question about characterisation of fundamental groups of smooth compact Kähler manifolds. Restrictions imposed by Kählerness are somewhat fragile, and if we ...
5
votes
1
answer
295
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What is the largest subgroup of $GL^{+}(7,\mathbb{R})$ which smoothly retracts onto $G_2$?
There is a nice smooth retraction from $\operatorname{GL}(n,\mathbb{C})$ onto $\operatorname{U}(n)$, which can be explained using polar decomposition. There is an analogous one from $\operatorname{GL}(...
4
votes
0
answers
207
views
Model structure on spaces with local coefficients
Is there a model structure on the category of topological spaces equipped with a local system (i.e. a functor from the fundamental groupoid to the category of abelian groups), such that the weak ...
3
votes
0
answers
316
views
Cubical approximation theorem for cubical complexes
A version of the simplicial approximation theorem states that a continuous map between finite simplicial complexes is homotopic to a simplicial map after subdividing the domain.
I have found a claim ...
2
votes
1
answer
596
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Triangulation induces regular CW complex structure
If a topological set is triangulable, dose the triangulation map gives it the (regular) CW complex structure? From definitions, I see it seems to be, but I am not that sure, for may exist some strange ...
4
votes
0
answers
111
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The fiber of the alternating map $X^{2n}\to \mathbb{Z}[X]$
Let $X$ be a fibrant connected simplicial set. There is a simplicial map $h_n\colon X^{2n}\to \mathbb{Z}[X]$, defined on points by $(x_1, \ldots x_{2n})\mapsto \sum\limits_{i=1}^{2n}(-1)^ix_i$. Here
$...
2
votes
0
answers
195
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Hypersurface containing nondegenerate subvariety of same degree and large dimension
Can a smooth hypersurface $X\subset \mathbb{P}_{\mathbb{C}}^{n+1}$ of degree $d$ contain a nondegenerate variety $Z$ with $\dim(Z)>\frac{n}{2}$ of degree $d$?
(If $r$ is the codimension of $Z$ in ...
8
votes
2
answers
384
views
Boundary triangulation induces triangulation
In $R^n$ (the real space) we have an open connected set $D$, such that $\partial D$ is triangulable. Can we prove the closure $\bar{D}$ is triangulable or any counterexample?
Furthermore, the $\...
122
votes
7
answers
15k
views
Topology and the 2016 Nobel Prize in Physics
I was very happy to learn that the work which led to the award of the 2016 Nobel Prize in Physics (shared between David J. Thouless, F. Duncan M. Haldane and J. Michael Kosterlitz) uses Topology. In ...
5
votes
1
answer
304
views
Simply connected manifolds with dense geodesics on the tangent bundle
A unit speed geodesic $\gamma:I\to M$ on a Riemannian manifold $M$ can be lifted to a curve $\sigma=(\gamma,\gamma')$ on $SM$, the unit sphere bundle (the unit tangent bundle) of $M$.
Let us say that ...
1
vote
0
answers
168
views
Why is any one Wirtinger relation a consequence of the remainder? [closed]
As the question title suggests, how do I see that any one Wirtinger relation is a consequence of the remainder?
5
votes
2
answers
559
views
Group of units of a ring spectrum vs of its connective cover
Let $R$ be a commutative ring spectrum (interpret this as you will; as an $E_\infty$-ring or as a commutative $S$-algebra etc.) and $\operatorname{GL}_1(R)$ as usual denote its space of units. If $\...
4
votes
0
answers
181
views
Smash Product of Frobenius Algebras
We have a smash product of Hopf algebras if one acts on other (namely making it module algebra, coalgebra and Hopf algebra) with a compatibility condition (Theorem 17).
Now I ask the same question ...
10
votes
0
answers
246
views
$[K(\mathbb Z,4),\mathbb H\text{P}^{\infty}]$
The map $\mathbb H\text{P}^{\infty} \to K(\mathbb Z,4)$ representing a generator of $H^4(\mathbb H\text{P}^{\infty};\mathbb Z) = \mathbb Z$ is a rational equivalence.
But is there any honest map in ...
7
votes
1
answer
440
views
Cohomology of the mapping class group of a non-orientable surface?
What is the low degree cohomology of the mapping class group of a non-orientable surface? More specifically, what is the universal central extension of the mapping class group of a non-orientable ...
2
votes
0
answers
206
views
Cohomology of fiber bundles with non constant coefficients
Let $E \to B$ be a topological fiber bundle. We know that the Serre spectral sequence allows you to compute the cohomology of $E$ with constant coefficient in terms of the cohomology of $F$ and the ...
4
votes
1
answer
410
views
Atiyah-Hirzebruch spectral sequence for a special kind of CW-complexes
Let $X$ be a finite CW-complex such that its $K$-theory $K^*(X)$ is, as a $\mathbb{Z}$-algebra, generated by $a_1, \cdots, a_n$ which are represented by reduced line bundles $L_1-1, \cdots, L_n-1$ ...
9
votes
2
answers
827
views
What is this analogy between manifolds and bundles (or schemes and locally free sheaves)?
There's a kind of analogy between the way manifolds work and the way bundles work. Let me try to give some examples of the analogy (although there may be better ones). I'll stick to smooth manifolds ...
5
votes
1
answer
229
views
$\pi_1$ of 4-manifolds that "look like" disk bundles
Let $X$ be a smooth compact oriented 4-manifold with $\partial X=L(p,1)$, $H_2(X;\Bbb Z)=\Bbb Z$, $H_3(X; \Bbb Z)=0$ and the induced map $\pi_1(L(p,1)) \to X$ surjective. What are the possibilities ...
7
votes
2
answers
375
views
are there finite nonabelian characteristic quotients $G$ of $F_2$ inducing a surjection $Aut(F_2)\twoheadrightarrow Aut(G)$?
Let $F_2$ be the free group of rank 2. Let $K\le F_2$ be a characteristic subgroup, such that $G := F_2/K$ is finite.
Do there exist examples of such nonabelian $G$ such that the induced map
$$Aut(...
3
votes
1
answer
529
views
Can lens spaces be realized by surgery along torus links?
As we know, the Lens space $L(p,q)$ is the quotient of $S^3$ by a $\mathbb{Z}_p$ action:
$(z_1,z_2) \rightarrow (e^{2\pi i/p}z_1,e^{2\pi iq/p}z_2)$.
It seems that the Lens space $L(1,0)$, a.k.a $S^3$,...
-2
votes
1
answer
247
views
Any galois covering of $P^{1}$ over rationals are of the form $\mathbb{P}^1_L\to\mathbb{P}^1_\mathbb{Q}$
I recently came across the following statement,
The Galois coverings of $\mathbb{P}^1_\mathbb{Q}$ are all of the form
$$\mathbb{P}^1_L\to\mathbb{P}^1_\mathbb{Q}$$ where $L$ is a number field.
How ...
-1
votes
1
answer
259
views
Question related to Galois covering of Projective line over rational numbers
Suppose that we have Galois covering $$X\longrightarrow P^{1}_{\mathbb{Q}}$$ defined over the rationals which is cyclic, in the sense that the Galois group associated to the covering is a cyclic group....
4
votes
0
answers
112
views
Reference request: Basic H-Space properties of $SO(3)$
I dug into the literature but could not find references for some of the basic H-space properties of $SO(3)$. Basic properties that I am looking for include
What H-maps are there $SO(3)\rightarrow S^3$...
10
votes
3
answers
620
views
Spin 4-manifold bounded by a mapping torus of tori
Consider a smooth torus endowed with the non-bounding spin structure. Pick a basis of its first homology and a diffeomorphism inducing the S-transformation
$\left(\begin{array}{cc} 0 & 1 \\-1 &...
4
votes
1
answer
266
views
Equivalence of two pictures of odd $K$-theory
One can show that two functors $K^0$ and $K_0(C(-))$ from the category of compact topological spaces to the category of abelian groups are naturally equivalent. The first one is topological $K$-theory ...
4
votes
2
answers
769
views
Homotopy pullbacks/relative homotopy groups vs homotopy pushouts/relative homology groups
In Goodwillie's "Calculus I", speaking of a commutative diagram of spaces
$$\begin{array}{c} Y & \rightarrow & Y_1 \\ \downarrow & & \downarrow & \\ Y_2 & \rightarrow & ...
1
vote
1
answer
171
views
Need help maximizing distances to nearest neighbor in a cylinder
I have a cylinder and I want to maximize the number of points in the cylinder such that the distances to the nearest neighbors are maximally spaced. How do I find out how many points I can have so ...
13
votes
1
answer
648
views
The fifth k-invariant of BSO(3)
From work of Pontryagin and Whitney, as I understand it, the homotopy 4-type of $BSO(3)$ is $K(\mathbb{Z}/2,2) \times_{K(\mathbb{Z}/4,4)} K(\mathbb{Z},4)$, where the pullback is along the maps $\...
2
votes
0
answers
933
views
Lifting of group homomorphisms
I asked this question a few days ago on math stackexchange but didn't get any answer so I thought I post it here too (see here):
In my first course on algebraic topology I heard about the following:
...
7
votes
1
answer
494
views
$G$ cocycle split to a coboundary in $J$, via a group extension
Consider a generic nontrivial $d$-cocycle $\omega_d^G \in H^d(G,U(1))$ in the cohomology group of a group $G$ with $U(1)=\mathbb{R}/\mathbb{Z}$ coefficient. In otherwords, here the $d$-cocycle $\...
2
votes
1
answer
264
views
Converse to the Weyl's theorem
Consider the following properties of a compact connected Lie group $G$:
(a) $G$ is semi-simple,
(b) $G$ has a finite fundamental group.
The well known Weyl's theorem states that (a) implies (b).
...
5
votes
1
answer
741
views
Topological Euler number of a singular variety
Let $X$ be a projective variety over $\mathbb{C}$. Is there a way to define some number $\tilde{\chi}(X)\in \mathbb{Z}$ satisfying both of the following two properties?
$\boldsymbol{(1)} \;$ When $X$ ...