Questions tagged [at.algebraic-topology]
Homotopy theory, homological algebra, algebraic treatments of manifolds.
8,259
questions
3
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0
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321
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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 ...
- 985
2
votes
1
answer
607
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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 ...
- 103
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
$...
- 493
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 ...
- 1,537
8
votes
2
answers
386
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 $\...
- 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
306
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 ...
- 8,046
1
vote
0
answers
168
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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?
- 19
5
votes
2
answers
564
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 $\...
- 1,991
4
votes
0
answers
182
views
Smash Product of Frobenius Algebras
We have a smash product of Hopf algebras if one acts on other (namely making it module algebra, coalgebra and Hopf algebra) with a compatibility condition (Theorem 17).
Now I ask the same question ...
- 191
10
votes
0
answers
246
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$[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 ...
- 11.9k
7
votes
1
answer
441
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 ...
- 12.4k
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 ...
- 121
4
votes
1
answer
417
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 $L_1-1, \cdots, L_n-1$ ...
- 383
9
votes
2
answers
832
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 ...
- 61.6k
5
votes
1
answer
229
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$\pi_1$ 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 ...
- 773
7
votes
2
answers
375
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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
$$Aut(...
- 7,161
3
votes
1
answer
530
views
Can lens spaces be realized by surgery along torus links?
As we know, the Lens space $L(p,q)$ is the quotient of $S^3$ by a $\mathbb{Z}_p$ action:
$(z_1,z_2) \rightarrow (e^{2\pi i/p}z_1,e^{2\pi iq/p}z_2)$.
It seems that the Lens space $L(1,0)$, a.k.a $S^3$,...
- 133
-2
votes
1
answer
247
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Any galois covering of $P^{1}$ over rationals are of the form $\mathbb{P}^1_L\to\mathbb{P}^1_\mathbb{Q}$
I recently came across the following statement,
The Galois coverings of $\mathbb{P}^1_\mathbb{Q}$ are all of the form
$$\mathbb{P}^1_L\to\mathbb{P}^1_\mathbb{Q}$$ where $L$ is a number field.
How ...
- 561
-1
votes
1
answer
259
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Question related to Galois covering of Projective line over rational numbers
Suppose that we have Galois covering $$X\longrightarrow P^{1}_{\mathbb{Q}}$$ defined over the rationals which is cyclic, in the sense that the Galois group associated to the covering is a cyclic group....
- 561
4
votes
0
answers
112
views
Reference request: Basic H-Space properties of $SO(3)$
I dug into the literature but could not find references for some of the basic H-space properties of $SO(3)$. Basic properties that I am looking for include
What H-maps are there $SO(3)\rightarrow S^3$...
- 5,016
10
votes
3
answers
622
views
Spin 4-manifold bounded by a mapping torus of tori
Consider a smooth torus endowed with the non-bounding spin structure. Pick a basis of its first homology and a diffeomorphism inducing the S-transformation
$\left(\begin{array}{cc} 0 & 1 \\-1 &...
- 1,582
5
votes
1
answer
268
views
Equivalence of two pictures of odd $K$-theory
One can show that two functors $K^0$ and $K_0(C(-))$ from the category of compact topological spaces to the category of abelian groups are naturally equivalent. The first one is topological $K$-theory ...
- 9,150
4
votes
2
answers
776
views
Homotopy pullbacks/relative homotopy groups vs homotopy pushouts/relative homology groups
In Goodwillie's "Calculus I", speaking of a commutative diagram of spaces
$$\begin{array}{c} Y & \rightarrow & Y_1 \\ \downarrow & & \downarrow & \\ Y_2 & \rightarrow & ...
- 2,302
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 ...
- 11
13
votes
1
answer
655
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 $\...
- 33.9k
2
votes
0
answers
938
views
Lifting of group homomorphisms
I asked this question a few days ago on math stackexchange but didn't get any answer so I thought I post it here too (see here):
In my first course on algebraic topology I heard about the following:
...
- 701
7
votes
1
answer
495
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$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 $\...
- 10.3k
2
votes
1
answer
264
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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
746
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 $X$ ...
- 1,049
1
vote
1
answer
397
views
Simple closed curves on genus 2 surfaces
Consider the family of proper complex genus two curves with affine equation $y^2 = x(x-1)(x-a)(x-b)(x-c)$, defined over an open subset $U$ of $\mathbb{C}^3$. Here, $U$ consists of all triples $(a,b,c)...
- 19
6
votes
1
answer
304
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Is being an NDR a local property?
I've asked this on MathSE without success:
https://math.stackexchange.com/questions/1929559/is-being-an-ndr-a-local-property
A pair of topological spaces $(X,A)$ is an NDR (neighborhood deformation ...
- 175
8
votes
0
answers
351
views
Tornehave's preprint "On BSG and the symmetric groups"
There are a few papers that cite Tornehave's preprint entitled On BSG and the symmetric groups apparently dating from early 70s or late 60s. Google search reveals very little. Does anyone have access ...
- 7,606
1
vote
0
answers
202
views
Simplicial approximation theorem and related stuff
I would like to ask for a list of references that explain the simplicial approximation theorem, barycentric subdivisions and Kan $Ex^{\infty}$ functor in details, fully exploiting the language of ...
- 1,403
1
vote
0
answers
290
views
Steenrod's geometric view of fiber bundles
In Dan Freed's notes pp.4 beneath (12.24) he comments that, let $P \to M$ be a principal $G$-bundle and $E = (P \times F)/G = P \times_G F$ its associated fiber bundle. The fibers of $E \to M$ are ...
- 447
0
votes
1
answer
249
views
Homotopy class of maps from torus to BG for a group G [closed]
I find a statement that the set of pairs of commuting elements in a group G is bijective to the set of homotopy classes of maps from torus to BG, the classifying space in the paper Elliptic cohomology ...
- 271
8
votes
2
answers
754
views
Models for equivariant genuine commutative ring spectra
The following question is an attempt at understanding various flavours of equivariant commutative ring spectra; it may not be suitable level for this forum.
Let $\mathcal{C}(G)$ be a symmetric ...
- 1,951
2
votes
1
answer
573
views
The concept "opposite" of Cohen-Macaulayness
Let $S = \mathbb k[x_1, \ldots, x_n]$ be a polynomial ring over a field $\mathbb k$ and let $I \subseteq S$ be a monomial ideal.
For a monomial ideal $J$, let $\#(J)$ be the smallest number of ...
- 449
11
votes
1
answer
430
views
Identification of a Serre Spectral Seq. via Thom Isomorphism with the Atiyah-Hirzebruch Spectral Seq
Let $\xi_n$ be an orientable $n$-dimensional vector bundle over a pointed space $B_n$. We can consider the relative Serre Spectral Sequence $$ H_p(B_n; h_q(D(\xi_n|\ast),S(\xi_n|\ast))\Rightarrow h_{p+...
- 2,018
8
votes
1
answer
413
views
Can we algorithmically contract loops in a simply connected space?
It is well-known that the question whether a given connected simplicial complex (or simplicial set) is simply connected, is algorithmically undecidable as it can model the word problem.
Assuming ...
- 962
7
votes
2
answers
704
views
Vector bundle over an oriented manifold with non-vanishing w_2w_3
I am looking for an example of an oriented rank 5 (or lower) real vector bundle $V$ over an oriented manifold such that the cup product $w_2(V) w_3(V)$ of Stiefel-Whitney classes does not vanish. It ...
- 1,582
14
votes
1
answer
658
views
What is the first Pontryagin class of the $n$-dimensional representation of $S_n$?
The symmetric group $S_n$ has an $n$-dimensional defining representation, which splits as $n = (n-1) + 1$. Although this representation exists integrally, I would like to think of this as a real ...
- 53.1k
1
vote
1
answer
381
views
Fiber bundle over a symplectic manifold
Let $(M,\omega)$ be a symplectic manifold. Does a vector fiber bundle $E$ exist, such that the Chern classes of $E$ are linked to the symplectic form as, for example, $\forall k$, $ch_k(E)=[\omega^k]$?...
- 377
1
vote
0
answers
133
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Standard proof that cyclic ordering of edges is preserved under planar graph homotopy?
I have several questions about the following theorem statement:
Thm: Let $G = (V, E)$ be a planar graph, and let $\varphi_0 : G \rightarrow \mathbb{R}^2$,
$\varphi_1 : G \rightarrow \mathbb{R}^2$ be ...
- 11
16
votes
1
answer
923
views
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 ...
- 27.2k
11
votes
2
answers
635
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$G$ cocycle split and trivialized to a coboundary in $J$, given a group homomorphism $J \overset{r}{\rightarrow} G$
Consider a generic nontrivial 3-cocycle $\omega_3^G(g_1,g_2,g_3) \in H^3(G,U(1))$ in the cohomology group of $G$ with $U(1)=\mathbb{R}/\mathbb{Z}$ coefficient. In otherwords, here the 3-cocycle $\...
- 10.3k
26
votes
1
answer
3k
views
Two points of view about Borel-moore homology
They are several ways to define the Borel-Moore homology on a locally compact space $X$.
The first one is by analogy with the singular homology but instead of using finite chains, we use locally ...
- 997
4
votes
1
answer
323
views
Finite second homotopy group
I am interested in sufficient conditions for the second homotopy group $\pi_2(X)$ of a compact connected manifold to be finite. Are there familiar classes of manifolds $X$, for which this is the case?
7
votes
1
answer
616
views
Explicit 2-cocycle from a 2nd cohomology group $H^2[Q_8 \times \mathbb{Z}/2\mathbb{Z}, U(1)]$
I would like to know the explicit expression of 2-cocycle from a 2nd cohomology group $H^2[Q_8 \times \mathbb{Z}/2\mathbb{Z}, U(1)]$ with $U(1)\equiv \mathbb{R}/\mathbb{Z}$ coefficient, or namely $H^2[...
- 10.3k
3
votes
0
answers
266
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Is there a spectral sequence for borel-moore homology associated to a whitney filtration?
Consider a Whitney stratified space
$$
\varnothing = X_{-1} \subseteq X_0 \subseteq X_1 \subseteq \cdots \subseteq X_n
$$
is there a spectral sequence for borel-moore homology which depends on the ...
- 1,706