All Questions
9,056 questions
149
votes
7
answers
23k
views
Homotopy groups of Lie groups
Several times I've heard the claim that any Lie group $G$ has trivial second fundamental group $\pi_2(G)$, but I have never actually come across a proof of this fact. Is there a nice argument, ...
- 4,014
5
votes
2
answers
714
views
On the boundary of a simply connected set
Let $U$ be an open simply connected subset of $\mathbb R^2$. Let $x$ be a boundary point of $U$.
Does then there always exist a continuous function $f\colon[0,1]\to\mathbb R^2\setminus U$ such that $x ...
- 128k
2
votes
0
answers
150
views
Local-to-global philosophy for crossed modules
In the survey Groupoids and crossed objects in algebraic topology Ronald Brown made after Corollary 5.17 (p 30) an very interesting remark I not fully understand. He stated that this
Corollary 5.17 ...
- 6,038
2
votes
0
answers
91
views
A recursive description of the smallest divisor-closed subsemigroup containing a set
Let $S$ be a semigroup and $\widehat{S}$ be its unitization, i.e., the monoid obtained from $S$ by adjoining an identity element if necessary (so that $\widehat{S} = S$ when $S$ is already a monoid).
...
- 10.5k
3
votes
1
answer
233
views
On infinity-morphisms between algebras over algebraic operads
I posted this question in the "Mathematics" stack exchange, but it hasn't got much attention... I hope it will get more here.
Let $P$ be a Koszul operad.
In the book of Loday-Vallette "...
- 215
13
votes
1
answer
386
views
Realizing integral homology classes on non-orientable manifolds by embedded orientable submanifolds
Let $M^m$ denote a compact, non-orientable smooth manifold and $\nu$ an integral homology class of dimension $n$. I am interested in understanding the representability of $\nu$ by embedded, orientable ...
- 587
6
votes
0
answers
144
views
Applications of $RO(G)$-graded computations outside of equivariant homotopy theory
While writing a grant proposal I faced a problem of justification my area of interest to a broader audience. So I thought it would be nice to ask it here:
What are applications/impact of computations ...
- 1,759
0
votes
1
answer
327
views
Can we generalise groupoids to monoid-oids? [closed]
Groups correspond to one object categories where every morphism is an isomorphism. Monoids correspond to one object categories.
Groupoids correspond to small categories where every morphism is an ...
4
votes
2
answers
342
views
On a generalized homotopy transfer theorem
In the book of Loday and Vallette "Algebraic Operads" a necessary condition for the Homotopy Transfer Theorem is that the starting operad is Koszul. I am interested in a generalization of ...
- 215
5
votes
0
answers
374
views
Comparing notions related to $(\infty,2)$-categories
I am trying to understand two related notions:
$(\infty,2)$-category as in Definition 5.5.1.3, Kerodon
weak $\infty$-bicategory as in Definition 4.1.1 in "$(\infty,2)$-Categories and the ...
6
votes
1
answer
202
views
Odd integral Stiefel–Whitney classes in terms of even ones
As computed in various places (e.g. in Brown - The Cohomology of $B\mathrm{SO}_n$ and $B\mathrm O_n$ with Integer Coefficients), the integral cohomology ring of $B\mathrm{O}(n)$ (and similarly $B\...
- 9,853
5
votes
0
answers
152
views
Group homology of $\mathrm{GL}_2(\mathbb{R})$ with real coefficients
What is known about the group homology of $\mathrm{GL}_2(\mathbb{R})$ with real coefficients and what are strategies to compute it (or at least some groups for low degrees)? Here I want to consider ...
- 1,174
6
votes
1
answer
457
views
Which maps of topological spaces have the right lifting property with respect to all split monomorphisms?
Let $p : X \to Y$ be a continuous map. We say that $p$ has the right lifting property with respect to split monomorphisms if, for every space $B$, and every retract $A \subseteq B$, and for every ...
- 64k
7
votes
1
answer
473
views
Why do homotopy orbits commute with the Eilenberg–Mac Lane spectrum functor?
Let $H: \mathsf{sAb}\to H\mathbb{Z}\text{-}\mathsf{Mod}$ denote the Eilenberg–Mac Lane functor sending a simplicial abelian group $M_\bullet$ to the infinite delooping of its geometric realization, i....
4
votes
1
answer
210
views
A question about Gysin exact sequence for cohomology of the orbit space
There is a claim in the following thesis regarding the exact sequence of Gysin. Shouldn't the spherical bundle $\mathbb{S}^1 \rightarrow X \rightarrow X/\mathbb{S}^1$ be orientable for the Gysin exact ...
- 1,367
1
vote
0
answers
133
views
Stable equivalence and stability theorem of vector bundles
I am going through this paper by Tanaka. In the proof of Proposition 3.2(1) given below
The author says that by the stability theorem as $\dim (B)\le m$ we have $\alpha\oplus1\cong m\oplus1$. But I ...
4
votes
2
answers
408
views
Classifying space of a non-discrete group and relationship between group homology and topological homology of Lie groups
I have a very soft question which might be very standard in textbooks or literature but I haven't seen it.
To a fixed group $G$ we may attach different topologies to make it different topological ...
- 441
2
votes
1
answer
165
views
string bordism group and framed bordism group for $d \leq 6$ and $d \geq 7$
Why do the string bordism group and the framed bordism group
coincide the same in dimensions lower than 7 ($d = 0,1,2,3,4,5, 6$)?
Why do the string bordism group and the framed bordism group differ
...
- 10.5k
5
votes
0
answers
159
views
Representing some odd multiples of integral homology classes by embedded submanifolds
Consider an $m$-dimensional compact closed orientable smooth manifold $M$ and an $n$-dimensional integral homology class $[\Sigma]$ on $M$, with $1 \le n \le m-1$. Then does there exist an odd integer ...
- 587
15
votes
6
answers
2k
views
Any shortcuts to understanding the properties of the Riemannian manifolds which are used in the books on algebraic topology
I'm now attending a reading seminar on the algebraic topology.
The seminar treats the book of Bott & Tu (Differential Forms in Algebraic Topology) and Milnor (Characteristic Classes).
In those ...
- 1,013
3
votes
1
answer
273
views
Do objects in the derived category behave stackily?
It is well known that derived categories (I'm particularly thinking of constructible derived categories and derived categories of D-modules) don't form a stack. In particular given morphisms in the ...
- 85
7
votes
0
answers
194
views
Are the spaces BG for compact connected groups G ind-projective or ind-Kähler?
Let $G$ be a compact connected group, or maybe better its complexification. By thinking about the simplicial Borel space, or using $n$-acyclic $G$-spaces for higher and higher $n$, it's "easy&...
- 44.7k
4
votes
0
answers
202
views
Possible Euler characteristics of manifolds with tangential structures
Let $p:B\to BO$ be a fibration. We say that a manifold has a $B$-structure if its stable tangent bundle lifts to $B$. I am interested in the question of whether there exists, for a given even ...
2
votes
1
answer
179
views
On the existence, for $\langle X,R\rangle$ a finite presentation of a group $G$, of an exact sequence of $\mathbb{Z}G$ modules
From this Q&A -- for $\langle X,R\rangle$ a finite presentation of a group $G$, there is an exact sequence of $\mathbb{Z}G$ modules
$$0\rightarrow\pi_{2}(Z)\rightarrow \mathbb{Z}G^{\oplus R}\...
21
votes
1
answer
638
views
Grothendieck group of the Fibonacci monoid
Let's denote the Fibonacci numbers by $F_0=0,F_1=1,F_{n+2}=F_{n+1}+F_n \; \forall n \ge 0$. According to Zeckendorf's theorem, every positive integer can be represented uniquely as the sum of some (at ...
- 1,543
1
vote
0
answers
168
views
Does the symmetric algebra functor preserve inclusions?
Theorem: For any compact abelian group $G$, the homogeneous component $%
H^{2}\left( B_{G};%
%TCIMACRO{\U{2124} }%
%BeginExpansion
\mathbb{Z}
%EndExpansion
\right) $ of degree $2$ is naturally ...
- 1,367
6
votes
0
answers
181
views
Extending topological vector bundles and obstruction theory
This is a question that has appeared in various forms on MathOverflow, see here and here, for example. But as opposed to these more algebraic questions, I am interested in the purely topological ...
- 1,763
5
votes
2
answers
479
views
Generalization of the concept of a measure
Consider the following generalization of the concept of a measure:
Let $L = (X, \lor, \land, \bot)$ be a semi-bounded lattice.
Let $M = (Y, \bullet, e)$ be a commutative monoid.
An $(L, M)$-measure is ...
- 2,203
4
votes
1
answer
476
views
Equivariant K-theory for products of groups?
Let $X$ be a $(G \times H)$-space. What is known about the connection between the groups $K_G(X)$, $K_H(X)$ and $K_{G \times H}(X)$? The $G$ and $H$ action on $X$ come from the canonical inclusions $G ...
- 293
3
votes
1
answer
203
views
Simple closed curves in a simply connected domain
Let $U$ be a bounded simply connected domain in the plane. Let $K$ be the boundary (or frontier) of $U$. For every $\varepsilon>0$ is there a simple closed curve $S\subset U$ such that the ...
- 3,317
231
votes
4
answers
16k
views
Is $\mathbb R^3$ the square of some topological space?
The other day, I was idly considering when a topological space has a square root. That is, what spaces are homeomorphic to $X \times X$ for some space $X$. $\mathbb{R}$ is not such a space: If $X \...
- 5,275
8
votes
3
answers
1k
views
Why did Ravenel define a ring spectrum to be flat if its smash-square splits into copies of itself?
In appendix A.2 of the orange book, Ravenel defines a ring spectrum $E$ to be flat if $E\wedge E$ is equivalent to a coproduct of suspensions of $E$. (Call this definition (1).) I've seen this ...
- 1,969
11
votes
1
answer
331
views
Embedded 2-tori in $S^1\times S^4$
I am interested in understanding the smooth isotopy class of embedded 2-tori in $S^1\times S^4$. Is it true that every two homotopic embedded 2-tori in $S^1\times S^4$ are smoothly isotopic? It would ...
- 129
8
votes
1
answer
470
views
Finite domination and compact ENRs
Edit: In the comments, Tyrone points out that West's positive answer to Borsuk's conjecture implies that every compact ENR is homotopy equivalent to a finite CW complex. It follows that the only ...
- 18.8k
2
votes
1
answer
296
views
Projective objects in chain complexes of an abelian category: Further question
Yes, I see there are other Q&A's on this, for instance here: Projective objects in the category of chain complexes
I am wondering why a level-wise projective chain complex $P$ which is split ...
- 511
1
vote
1
answer
270
views
Cohomological dimension of kernel
Let $M$ and $N$ be two connected, smooth (possibly non-compact), aspherical manifolds (that is, $\pi_k(-)=1$ for $k\geq 2$) of dimensions $n$ and $n-r$ respectively. Let $f:M\to N$ be a smooth map ...
- 585
6
votes
1
answer
315
views
Commuting homotopy colimits and arbitrary products in spaces
Let $X : D \rightarrow Spc$ be a diagram with values in the $\infty$-category of spaces and $I$ some (discrete) set, not necessarily finite. ($D$ can be a 1-category if that makes statements easier, ...
- 2,303
6
votes
1
answer
375
views
Different flavours of Vassiliev Conjecture
There is something that puzzles me about "Vassiliev's Conjecture". I am sure I am missing some detail which is obvious to the community, since there are several tightly related kind of ...
- 2,152
11
votes
0
answers
225
views
The algebras and coalgebras of the homology functor
My question is very simple, but I suspect far from the intuition with which singular homology is introduced.
Consider singular homology as a functor
$$H_n : {\sf Top}\times{\sf Ab} \to \sf Ab$$
This ...
- 13.6k
5
votes
1
answer
215
views
Aspherical subcomplexes of finite aspherical 2-complexes
Recently I have been considering the following question in low-dimensional topology, that would help with another result I have been working on, but have been assuming that the statement is likely too ...
- 187
20
votes
3
answers
3k
views
How can Machine Learning help “see” in higher dimensions?
The news that DeepMind had helped mathematicians in research (one in representation theory, and one in knot theory) certainly got many thinking, what other projects could AI help us with? See MO ...
- 485
12
votes
2
answers
1k
views
Why are ordinary spheres not strictly invertible?
Introduction
This question is about Picard spectra for the symmetric monoidal $\infty$-category of spectra. We say that a spectrum $X$ is invertible if there is another spectrum $Y$ such that $X\...
- 56.9k
1
vote
0
answers
153
views
Poincaré-Hopf Theorem for domains with a point of vanishing curvature
Consider $\Omega \subset \mathbb{R}^2$ a convex planar domain having positive curvature on the boundary except for a point $p \in \partial \Omega$ where the curvature vanishes.
I would like to know ...
- 111
10
votes
0
answers
248
views
What is the tiling semigroup for an einstein "hat" tiling?
My undergraduate dissertation was on inverse semigroups and the key text I used for it was Lawson's, "Inverse Semigroups: The Theory of Partial Symmetries". In said book, Lawson describes ...
- 379
2
votes
0
answers
68
views
cocycle datum for principal $G$-bundle over base space Delta set
Let $X$ be topological realization of a (finite)
Delta set, $G$ a finite group and $p: P \to X$ a
principal $G$-bundle.
Let's recall the standard fact that more generally any
numerable principal G-...
- 619
8
votes
2
answers
264
views
Is the fiberwise suspension of a Serre fibration a Serre fibration?
Let $\pi\colon X \rightarrow Y$ be a Serre fibration. Define $\Sigma_f\pi \colon \Sigma_f X \rightarrow Y$ be the fiberwise unreduced suspension of $\pi$. Thus $\Sigma_f X = X \times [0,1] / {\sim}$,...
- 83
8
votes
1
answer
757
views
The image of the J-homomorphism of the tangent bundle of the sphere
Consider the J-homomorphism $\pi_{n-1}(SO(n))\to \pi_{2n-1}(S^n)$ and $\tau_{S^n}:S^{n-1}\to SO(n)$ to be the adjoint of the classifying map of the tangent bundle of the standard sphere. In this paper,...
7
votes
0
answers
191
views
Complex cobordism and integrable systems
In Jack Morava's paper On the complex cobordism ring as a Fock representation, it was remarked right at the beginning that complex cobordism may play a role in the theory of integrable systems. In ...
- 71
2
votes
0
answers
198
views
Proposition 4.3.8 Qing Liu about flat morphisms of schemes
I have a problem with a detail of Qing Liu's proof of Proposition 4.3.8 (pag. 137 of "Algebraic Geometry and Arithmetic Curves").
The statement is:
Let $Y$ be a scheme having only a finite ...
- 149
12
votes
0
answers
492
views
Proofs of Serre's theorem on simply-connected finite CW complexes
A famous result due to Serre states that any simply-connected finite CW complex with non-trivial $\mathbb{Z}_2$ homology has infinitely many non-zero homotopy groups. (In fact, Serre proves more than ...
