All Questions
9,056 questions
3
votes
1
answer
508
views
Handlebody decomposition of $L(2,1)\times S^1$
I wish to know the handlebody decomposition of $L(2,1)\times S^1$ in terms of Kirby diagrams, where $L(2,1)\cong RP^3$. And if possible, is there a general recipe for getting the handlebody ...
- 125
17
votes
2
answers
1k
views
Homotopy groups of Diff(X) and Homeo(X)
For a compact closed smooth manifold $X$, the group Diff(X) has a natural homomorphism $\Phi$ to the homeomorphism group Homeo(X). If $X$ has dimension at least $5$, I'm looking for some general ...
- 19.4k
4
votes
0
answers
316
views
Loop-suspension of degree d map of sphere
Let $f\colon S^n \rightarrow S^n$ be a basepoint preserving map of degree $d \geq 1$. We then get an induced map $\Omega \Sigma(f)\colon \Omega \Sigma S^n \rightarrow \Omega \Sigma S^n$. There is ...
- 41
9
votes
1
answer
812
views
Is there a version of the Poincaré–Hopf theorem for manifold with corners?
As we know, the square $S=[0,1]\times[0,1]$ is not a manifold with boundary. Instead, it's a manifold with corners. For a tangent vector field on a compact manifold with boundary, we have the Poincaré–...
- 93
3
votes
1
answer
257
views
Why does this construction not give a functorial cone in the homotopy category of cochain complexes?
I have heard the expression recently that one should be careful when constructing cones in the homotopy category - namely, that this is not functorial. However, when working through some examples in ...
- 285
46
votes
2
answers
4k
views
What are the potential applications of perfectoid spaces to homotopy theory?
This year's Arizona Winter School was on perfectoid spaces, and there were quite a few homotopy theorists in the audience. I'd like to get a "big list" of reasons homotopy theorists might care about ...
7
votes
0
answers
295
views
A minimal semigroup generating subset of the additive reals
I asked this on MSE, but I was told to ask it here because it is a difficult question. Consider the additive magma of the real numbers, $(\mathbb{R};+)$. Does there exist a subset $S$ of the reals ...
- 2,023
11
votes
2
answers
1k
views
Effect of the curse of dimension on collision detection
I require a crude intersection analyzer or collision detector in about 15 dimensions. I am wondering if such a function is rendered difficult or impossible by the "curse of dimension". ...
- 119
15
votes
5
answers
2k
views
Striking existence theorems with mild conditions, and simple to state: more recent examples?
I would like to write an article about powerful existence theorems that assert, under mild and simple conditions, that some basic pattern or regularity exist. See some examples below. By mild ...
4
votes
1
answer
255
views
Closed good cover of a triangulable space
By a good closed cover of a topological space $X$, I mean a collection of closed subspaces of $X$, such that the interior of them cover $X$, and any finite intersection of these closed subspaces is ...
- 357
24
votes
10
answers
4k
views
Why localize spaces with respect to homology?
A basic construction in algebraic topology is the localization of spaces or spectra with respect to a homology theory: one formally inverts the $E$-homology isomorphisms, reflecting each space into ...
- 66.8k
3
votes
0
answers
258
views
Determinantal variety
It is well known in literature about the determinantal varieties, symmetric determinantal varities, skew-symmetric determinantal varieties. Is it possible to study determinantal varieties over the ...
- 31
6
votes
1
answer
374
views
Does the Lie algebra structure on rational homotopy groups reflect similar information to the formal group structure in characteristic p?
It's well known (c.f. Quillen and Sullivan) that the rational homotopy theory of spaces is equivalent to the homotopy theory of rational DG-algebras; in particular, rational spaces and rational ...
- 1,969
14
votes
2
answers
1k
views
Atiyah duality without reference to an embedding
Atiyah duality is the equivalence $M/\partial M \simeq (M^{-T(M)})^\vee$, i.e. the Spanier-Whitehead dual of the space $M/\partial M$ is the Thom complex of the stable normal bundle of $M$. The ...
- 5,849
3
votes
0
answers
127
views
Algebraic models of cohomology classes of (higher) Eilenberg-Maclane Spaces?
In Classification of weak 3-groups, Qiaochu gave an excellent answer, in which, he mentioned cohomology classes $H^{4}(B^{2}\pi_{2};\pi_{3})$ can be viewed as quadratic refinement of Whitehead bracket ...
- 93
0
votes
0
answers
215
views
Null-homotopicness of an inclusion map
Let $K$ and $L$ be simplicial complexes such that 1) $L\subseteq K$; 2) $K$ is homotopic to $S^4$; 3) $L$ is homotopic to $S^6$.
Is the inclusion map from $L$ to $K$ null-homotopic?
Thanks!
2
votes
1
answer
154
views
Do there exist smaller simplicial models of barycentric subdivisions?
Let $S$ be a simplicial complex and let $Bary(S)$ denote its barycentric subdivision.
Of course, the geometric realizations of $S$ and $Bary(S)$ are homeomorphic.
However, one issue that arises in ...
1
vote
0
answers
37
views
Unique smallest degree algebraic solution to polynomial ODE
Let's assume we are given a degree $d$ polynomial VF as a map $f: \mathbb{R}^n \rightarrow \mathbb{R}^n$
$$f_j(x)=\sum_{i_{1},\dots,i_{n}=1}^{d}a^j_{i_{1},\dots,i_{n}}x_{1}^{i_{1}}\dots x_{n}^{i_{n}}$$...
- 247
6
votes
1
answer
1k
views
Why is a simply connected homology sphere a topological sphere?
I post this for a friend who currently doesn’t have access to this site.
It is about an implication in the last paragraph of the following paper:
KATSUHIRO SHIOHAMA and HONGWEI XU, The topological ...
- 136
14
votes
2
answers
4k
views
Mistakes in Bredon's book "Topology and Geometry"?
I am preparing the notes for a course in Algebraic Topology, so I decided to borrow some of the material from the classical (and wonderful) book by G. Bredon Topology and Geometry.
Looking at the ...
- 66.3k
2
votes
0
answers
242
views
Monodromy group action on de Rham cohomology
Let $f : Y \longrightarrow X := \mathbb{P}^1\setminus\{0,1,\infty\}$ be the smooth proper morphism associated to the Legendre family, which is an elliptic fibration of the punctured line, with fibre ...
- 2,907
5
votes
1
answer
244
views
Coefficient of the top Pontryagin class in $L$-genus
The $L$ genus can be expressed as combinations of the Pontryagin classes with the first few terms as follows:
$$L_1=\frac{1}{3}p_1,$$
$$L_2=\frac{1}{45}(7p_2-p_1^2),$$
$$L_3=\frac{1}{945}(62p_3-...
- 707
6
votes
1
answer
812
views
Homotopy theory with condensed sets
Is there a canonical way for doing homotopy theory with condensed sets? Is there a definition of homotopy groups? As CW complexes are compactly generated Hausdorff we can consider then as condensed ...
- 1,485
7
votes
1
answer
278
views
What were the "questions unapproachable by other means" w.r.t. $KO$-invariants?
H. Blaine Lawson, Jr. and Marie-Louise Michelsohn, Spin Geometry, (1989), p. xi:
...This formula was to generalize the important [HRR]. ...Atiyah and Singer...produced a globally defined elliptic ...
- 609
15
votes
5
answers
2k
views
Reading list for Equivariant Cohomology
I was applying equivaraint cohomology, in particular in the symplectic setting, for some time, but I feel like I am missing some nice books/course notes/articles. Could you advise me some literature, ...
- 14.3k
4
votes
0
answers
191
views
Every closed surface divides a closed 3-manifold $X$ into two parts if and only if $H_1(X)$ is finite
My friend is looking for proof of the following statement
Every closed surface divides a closed 3-manifold $X$ into two parts if and only if $H_1(X; \mathbb{Z})$ is finite.
Rumor source: Justin ...
- 6,962
2
votes
1
answer
116
views
On the maximum elements of a numerical semigroup that have order between $n$ and $2n$
Let $S$ be a submonoid of the non-negative integers $\mathbb Z_{\geq 0}.$ If $\mathbb Z_{\geq 0} \setminus S$ is finite, we say that $S$ is a numerical semigroup. Let $S^*$ denote the collection of ...
- 183
49
votes
4
answers
7k
views
Elegant proof that any closed, oriented 3-manifold is the boundary of some oriented 4-manifold?
I'm looking for an elegant proof that any closed, oriented $3$-manifold $M$ is the boundary of some oriented $4$-manifold $B$.
- 1,709
6
votes
1
answer
483
views
Exit path categories of regular CW complexes
Given a finite, regular CW complex $X$ (by regular, I mean that the gluing maps $D^n \to X$ from the closed unit ball to $X$ are homeomorphisms onto their image), denote by $S$ the finite partially ...
- 898
9
votes
1
answer
743
views
Why is choice needed in Ellis' Lemma?
Ellis Lemma on idempotent elements asserts that:
Lemma (Ellis). Every compact semigroup has an idempotent.
The proof below is excerpted from Todorcevic's Introduction to Ramsey Spaces, Lemma 2.1.
...
- 1,442
3
votes
0
answers
198
views
When is $BG \rightarrow BH \rightarrow BK$ a principal fibration?
Let $1 \rightarrow G \rightarrow H \rightarrow K \rightarrow 1$ be a short exact sequence of groups. Assume for simplicity that $G$ is finite, with the discrete topology (so $BG$ is a $K(G,1)$). ...
- 371
31
votes
3
answers
1k
views
Non embedding of $Y\times Y$ into $\mathbb{R}^3$
I know that this is a well known result, but where can I find a proof? I am also interested to see more general non-embedding results of this type.
Theorem. Let $Y$ be the union of two segments ...
- 28k
1
vote
0
answers
54
views
The number of $n$-cells attaching to $K^{n-1}$ in Wall's construction
Let $\phi:K\to X$ be a map, with mapping cyliner $M=X\cup_{\phi}(K\times I)$. We define $\pi_n (f)$ as $\pi_n (M,K\times 1)$. An element of $\pi_n (f)$ is represented by a pair of maps $\beta :S^{n-1}\...
- 287
2
votes
1
answer
252
views
If $M$ is a compact smooth finite-dimensional manifold with boundary, is the inclusion of a closed subspace $A \subseteq M$ a cofibration?
Question: If $M$ is a compact smooth finite-dimensional manifold with boundary, is the inclusion of a closed subspace $A \subseteq M$ a cofibration? (I'm specifically interested in the case when $A$ ...
- 123
12
votes
3
answers
533
views
Small simplicial set models for BG
Let $F$ be a finite group.
Is there a model for $BF$ as a simplicial set such that the number of nondegenerate $n$-simplices grows at most polynomially?
For example the Bar construction has the ...
- 11.1k
2
votes
0
answers
68
views
Confusion about Turaev's description of G-bundles on the cylinder and pairs of pants
In Homotopy Field Theory in dimension 2 and group algebras, section 4.6, page 24, Turaev considers an annulus $C = S^1 \times [0,1]$ (thought of as a cobordism from $C_0 = S^1 \times \{ 0\}$ to $C_1 = ...
- 371
10
votes
2
answers
2k
views
Parallelizability of 3-manifolds
Robert Bryant's answer here ( https://mathoverflow.net/a/149496/85500 ) states that any orientable 3-manifold is parallelizable.
Previously I was under the impression that only closed (compact & ...
- 3,307
5
votes
2
answers
322
views
Can a punctured $\mathbb{C}P^n$ be a retract of a punctured $\mathbb{C}P^{n+1}$?
In this question we consider the standard inclusion of $\mathbb{C}P^{n}\subset \mathbb{C}P^{n+1}$.
It is well known that $\mathbb{C}P^n$ is not a retract of $\mathbb{C}P^{n+1}$.
What about if we ...
- 366
4
votes
1
answer
320
views
Higher order differentials of Bockstein spectral sequence
The Bockstein SS is obtained from the exact sequence
$$0\to\mathbb{Z}\xrightarrow{2}\mathbb{Z}\to\mathbb{Z}/2\to 0$$
with $E_1^p=H^p(X,\mathbb{Z}/2)$ and the differential $d_1=Sq^1$.
How to identify ...
- 918
23
votes
2
answers
467
views
Which homotopy 2-types are H-spaces?
Let $X$ be a connected CW-complex with $\pi_k(X)$ trivial for $k >2$. Is it known under which circumstances $X$ is an $H$-group?
I have been able only to deduce the necessary condition that $\pi_1(...
- 344
5
votes
1
answer
322
views
Is there a model structure for S-modules such that cofibrant operad-algebras forget to cofibrant S-modules?
In 1997, Elmendorf, Kriz, Mandell, and May wrote a book Rings, Modules, and Algebras in Stable Homotopy Theory in which they introduced the category of $S$-modules as a model for the stable homotopy ...
- 30.3k
6
votes
1
answer
332
views
Is every simplicial map $\Phi:K(A, n) \to K(A', n)$ a simplicial homomorphism of groups?
I have posted a few questions on MSE, most notably this one, which revolve around the same issue and have received no answers, so I decided to ask the same here.
In the following, $K(A, n)$ is the ...
- 265
12
votes
3
answers
950
views
Smooth map homotopic to Lie group homomorphism
Let $G$ and $H$ be connected Lie groups. A Lie group homomorphism $\rho:G\to H$ is a smooth map of manifolds which is also a group homomorphism.
Question: Can we find a smooth (or real-analytic) map $...
- 2,789
20
votes
5
answers
4k
views
Universal property of the smash product (of pointed spaces)
Is there a universal property for the smash product (of pointed spaces or pointed CW-complexes or something of that ilk)? I've seen the smash product of spectra defined with a universal property in ...
- 1,147
6
votes
2
answers
618
views
Classifying space $\text{BU}(n)$ from the differential-geometric point of view?
The classifying space of a topological group $G$ is the quotient of $EG$ (a topological space with vanishing homotopy groups) by a proper free action of $G$. The standard notation for the classifying ...
- 71
2
votes
0
answers
425
views
About infinite loop space and $\Omega$ spectrum
Let $A$ is an topological abelian monoid. Also $\pi_0(A)$ is a group and $A$ has $CW$ structure.
$BA$ is a classifying space of the topological abelian monoid.
My purpose is to construct an infinite ...
- 121
3
votes
0
answers
164
views
Spin structures on surfaces in terms of homology classes
It is well known that the spin structures on an oriented surface (with boundary) $M$ are in bijection with the set of cohomology classes $H^1(M,\mathbb{Z}/2)$. By Lefschetz duality, these correspond ...
- 371
3
votes
1
answer
171
views
Spaces satisfying a strong Cartan-Hadamard theorem
Let $(X,d)$ be a connected geodesic metric space. When does there there exists a covering map $\pi:H\rightarrow X$ which is a local-isometry where $H$ is either a Hilbert space or a Euclidean space?
...
- 364
2
votes
1
answer
267
views
on second cohomology of $S^1$-manifold
Let $M$ be a closed oriented real manifold with a free smooth circle action. Denote $BS^1$ to be the classifying space of principal circle bundles and $ES^1\rightarrow BS^1$ to be the universal ...
- 113
3
votes
0
answers
145
views
Formality of Sullivan Representatives
Suppose we have a map $f : \mathcal{A} \to \mathcal{B}$ between two formal, simply connected CDGAs, with induced map on cohomology $H(f) : H(\mathcal{A}) \to H(\mathcal{B})$. Further, suppose we have ...
- 285