Questions tagged [at.algebraic-topology]

Homotopy theory, homological algebra, algebraic treatments of manifolds.

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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}^...
Karol Szumiło's user avatar
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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 ...
Stanley Chang's user avatar
15 votes
1 answer
376 views

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 $\...
David C's user avatar
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6 votes
1 answer
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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 $C^\infty(M,N)$?
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9 votes
2 answers
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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?
user378415's user avatar
3 votes
1 answer
271 views

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 ...
user's user avatar
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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 $$\...
JoeB's user avatar
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4 votes
1 answer
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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 $...
Omar's user avatar
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1 answer
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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{...
Alexander Körschgen's user avatar
4 votes
0 answers
173 views

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 ...
Daniel Grady's user avatar
4 votes
1 answer
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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 ...
HeinrichD's user avatar
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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 ...
Ali Taghavi's user avatar
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 ...
Anton Kapustin's user avatar
16 votes
2 answers
2k views

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 ...
C.F.G's user avatar
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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),...
Nicolas Boerger's user avatar
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 ...
David Treumann's user avatar
6 votes
0 answers
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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 ...
user avatar
6 votes
1 answer
446 views

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 ...
Denis T's user avatar
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5 votes
1 answer
295 views

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}(...
Malkoun's user avatar
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4 votes
0 answers
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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 ...
local's user avatar
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3 votes
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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 ...
Ben Knudsen's user avatar
2 votes
1 answer
596 views

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 ...
lun zhang's user avatar
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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 $...
Sergey Sinchuk's user avatar
2 votes
0 answers
195 views

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 ...
DCT's user avatar
  • 1,537
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 $\...
lun zhang's user avatar
  • 103
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 ...
Joonas Ilmavirta's user avatar
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?
user99231's user avatar
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 $\...
A Rock and a Hard Place's user avatar
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 ...
Kadir Emir's user avatar
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 ...
Jens Reinhold's user avatar
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 ...
Kevin Walker's user avatar
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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 ...
cannonball's user avatar
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 $L_1-1, \cdots, L_n-1$ ...
No_way's user avatar
  • 383
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 ...
Tim Campion's user avatar
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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 ...
PVAL's user avatar
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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(...
stupid_question_bot's user avatar
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 $S^3$,...
Franklin Wu's user avatar
-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 ...
Tensor_Product's user avatar
-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....
Tensor_Product's user avatar
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 $SO(3)\rightarrow S^3$...
Tyrone's user avatar
  • 5,016
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 &...
Samuel Monnier's user avatar
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 ...
truebaran's user avatar
  • 9,140
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 & ...
Dimitri Chikhladze's user avatar
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 ...
user98725's user avatar
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 $\...
David Roberts's user avatar
  • 33.8k
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: ...
M.U.'s user avatar
  • 701
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 $\...
wonderich's user avatar
  • 10.3k
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). ...
William of Baskerville's user avatar
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 $X$ ...
user44651's user avatar
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