All Questions
8,725 questions
3
votes
1
answer
238
views
Steenrod operations on classifying spaces
Steenrod operations can be defined for all finite characteristics $p$. The simplest one, when $p=2$, is the Steenrod square. I wonder if the computation for classifying spaces for classical Lie groups ...
- 803
7
votes
0
answers
380
views
Reference request: cohomology of BTOP with mod $2^m$ coefficients
I am searching for a reference with information pertaining to the $\mathbb{Z}/{2^m}$ cohomology of ${\rm{BTOP}}(n)$, for $n \geq 8$ and $m=1,2$, where
$${\rm{TOP}}(n) = \{f \colon \mathbb{R}^n \to \...
- 371
7
votes
1
answer
292
views
Computing $\pi_1$ of the complement of a non-singular plane curve
The following is a well-known fact:
Theorem. The fundamental group of the complement of a non-singular curve of degree $d$ in the complex projective plane is cyclic of order $d$.
This was further ...
- 10.9k
4
votes
1
answer
430
views
Criteria for extending vector field on sphere to ball
Below is a theorem that is equivalent to Brouwer fixed-point theorem, which I found quite interesting. The proof is in this PDF file.
Let $v: \mathbb S^{n-1} \to \mathbb R^n$ be a continuous map, ...
- 807
3
votes
1
answer
134
views
Are the two families of Johnson invariants of the Torelli groups related beyond the first one?
$\newcommand{\sp}{\operatorname{Sp}(H)}$
$\newcommand{\gr}{\operatorname{gr}}$
$\newcommand{\id}{\operatorname{id}}$
$\newcommand{\der}{\operatorname{Der}}$
Johnson has defined two families $\tau_k,\...
- 8,524
3
votes
1
answer
134
views
Geodesic laminations on the 4-punctured sphere
Let $S_{0,4}$ be the 4-punctured sphere equipped with a hyperbolic metric of finite volume (thus all punctures are cusps). Consider $\gamma$ to be a simple geodesic of $S_{0,4}$ (not necessarily ...
- 133
5
votes
1
answer
249
views
Double hom with $\mathbb{CP}^\infty$
Pontrjagin duality gives a double dual theorem for "hom with $S^1$", and $S^1$ is $\textbf{B}\mathbb{Z}$ up to homotopy.
$\textbf{B} \textbf{B}\mathbb{Z}$, modeled by $\mathbb{C}\mathbb{P}^{\...
user30211
8
votes
1
answer
439
views
Model categories as a tool to resolve size issues for localizing categories
I have a rather basic question about one motivation for introducing model categories in David White's notes, as a possible way to overcome troubles with size issues appearing when localizing a ...
- 6,048
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
3
votes
1
answer
200
views
Does the group of compactly supported diffeomorphisms have the homotopy type of a CW complex?
It is known that the group of diffeomorphisms of a compact manifold with the natural $C^{\infty}$ topology has the homotopy type of a countable CW complex. See for instance this thread: Is the space ...
- 491
11
votes
1
answer
659
views
Can you deduce the correspondence between 2D oriented TQFTs and commutative Frobenius algebras from the (framed) Cobordism Hypothesis?
Background
I am currently writing an MSc dissertation on TQFTs (and Khovanov homology, but that is unrelated to this question).
After having read most of Kock's book on the equivalence between 2D ...
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
13
votes
1
answer
2k
views
Who proved the motivic 6-functor formalism?
In the recent beautiful talk "Motives and ring stacks" Peter Scholze states the theorem saying that there exists an initial 6-functor formalism on $\mathit{Sch}_\mathbb{Z}$ such that
when $...
- 705
5
votes
1
answer
139
views
Minimal cell structures in combinatorial model categories
I recently rediscovered a classical theorem from Hatcher which states that simply-connected CW complexes have a 'minimal' cell structure, where the cells correspond to spheres and disks indexed by the ...
- 285
3
votes
1
answer
151
views
Can a phantom map have finite cofiber?
Let $f : X \to Y$ be a nonzero phantom map between spectra. Can the cofiber of $f$ be a finite spectrum?
Recall that a map $f$ is said to be phantom if $f \circ i = 0$ whenever $i : F \to X$ is a map ...
- 64k
2
votes
1
answer
184
views
combinatorical description of classifying map for principal $G$-bundle over 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. It's standard that the isom' classes of such principal $G$-bundles are ...
- 619
3
votes
2
answers
364
views
Is the mapping cylinder a replacement for morphism by cofibration in model categories?
Let $M$ be a model category, consider a very good cylinder object $X \coprod X \to X \times I \overset{\operatorname{pr}}{\to} X$ (here $X \times I$ is just a notation, no object $I$ is implied), that ...
- 6,962
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 ...
5
votes
1
answer
270
views
Are Euclidean spaces $\Delta$-generated?
From the definition of $\Delta$-generated it seems like $\mathbb R$ should be $\Delta$-generated, as $\mathbb R$ is final with respect to all continuous maps $\mathbb R^n \to \mathbb R$.
However, the ...
- 53
3
votes
1
answer
137
views
Derived flat bundles
I am looking for a notion of derived flat bundles over a surface $X$. Flat vector bundles may be thought of in terms of surface representations $\pi_1(X)\rightarrow\text{GL}(V)$. Is there a notion of ...
- 31
3
votes
0
answers
50
views
Natural morphisms between stable unitary, orthogonal, and (compact) symplectic groups
I am a physicist knowing a bit of algebraic topology, and trying to answer the following question.
This is perhaps not appropriate as a question on MO, in which case I apologize.
I posted this ...
- 131
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 ...
- 6,048
1
vote
1
answer
237
views
Symmetric-monoidal-associative smash product up to homotopy
I am thinking about sequential spectra. I am trying to figure out if the smash product here is symmetric monoidal associative up to homotopy. See definition 3.16 in the above.
Recall that a sequential ...
user30211
2
votes
0
answers
208
views
Classification of bundles with fixed total space
I am aware of classification theorems for principal bundles, vector bundles, and covering spaces $\pi:E\to B$ over a fixed base space $B$. Principal and vector bundles over $B$ are classified by ...
- 501
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
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
13
votes
1
answer
518
views
Impossibility of realizing codimension 1 homology classes by embedded non-orientable hypersurfaces
Suppose we have an $n+1$-dimensional compact closed oriented manifold $M$ and an $n$-dimensional integral homology class $[\Sigma]\in H_n(M,\mathbb{Z})$ on $M.$ Then is it true that $[\Sigma]$ mod $2$ ...
- 587
7
votes
1
answer
2k
views
Which revolutions in topology and geometry can we expect in the next 20 years? [closed]
In my limited perspective on the history of mathematics, I can name at least two big revolutions in Topology and Geometry (broadly construed): the introduction of Schemes in Algebraic Geometry, and ...
2
votes
1
answer
230
views
On the definition of a derived $A_\infty$-category
Let $\mathcal{A}$ be an $A_\infty$-category. The derived $A_\infty$-category is defined to be the 0th cohomology category of the category of twisted complexes of $\mathcal{A}$.
I have troubles ...
- 219
2
votes
0
answers
190
views
Connection on relative topological periodic cyclic homology
I have been looking Bhatt-Morrow-Scholze's paper:
https://arxiv.org/pdf/1802.03261.pdf
and came to a naive question. Let $C$ be a dg-category (with assumptions?) over $\mathbb{F}_p[[z]]$ and view this ...
- 3,758
4
votes
0
answers
206
views
Reference for a folklore theorem about h-cobordisms
I've seen referenced here that if $M$ and $N$ are closed topological $n$-manifolds and $f: \mathbb{R}\times M \to \mathbb{R}\times N$ is a homeomorphism, then $M$ and $N$ are h-cobordant.
I know that ...
- 41
10
votes
1
answer
233
views
Classifying space of centralizer
$\DeclareMathOperator\Map{Map}\newcommand{\B}{\mathrm{B}}\newcommand{\h}{\mathrm{h}}$Let $f:G\to H$ be a morphism of topological groups and let
$$H^{\h G}:=\Map_G(\mathrm{E}G, H)$$
be the homotopy ...
- 103
0
votes
1
answer
327
views
what is this simple topological space?
Take $M_p$ the mapping cylinder (MC) of the $p=3$-fold cover of $S^1$, $M_q$ the MC of the $q=2$-fold cover of $S^1$, where for both, the identification of the MC is done on the side $\{0\}$ of $[0,1]$...
3
votes
1
answer
317
views
"Totally real" linear transformations
Identify $\mathbb{C}^n$ with $\mathbb{R}^{2n}$ via the equality $$(z_1, z_2, \ldots, z_n)=(x_1, \ldots, x_n, y_1, \ldots , y_n)$$
Where $z_j=x_j + iy_j$.
We call a linear invertible map $A: \mathbb{R}^...
user515519
24
votes
1
answer
1k
views
What topological principle is at work here?
[I'm cross-posting this from MSE. I initially asked there 10 days ago, and the question was well-received, but left unanswered.]
My question is inspired by a problem I discovered in Putnam and Beyond,...
- 956
4
votes
0
answers
249
views
Homotopy group of maps into S^3 using its Lie group multiplication to define the group operation
The Bruschlinsky group of maps of a space X into S1 up to homotopy, using the multiplication on S1, is well-known to equal the first cohomology group of X (at least assuming X is a reasonably nice ...
- 2,858
14
votes
1
answer
869
views
What is $\pi_{23}(S^2)$?
The first $22$ homotopy groups of the $2$-sphere were worked out by Toda in 1962, but I cannot find any results extending that to any higher homotopy groups of $S^2$.
Are any more of these groups ...
- 827
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 ...
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
3
votes
1
answer
134
views
Recover the $C_k$-action of a cyclic object as from the $S^1$-action on Hochschild chain
$\newcommand{\Fun}{\operatorname{Fun}}$Let $X\in\Fun(\Lambda^\mathrm{op}, \mathcal{C})$ be a cyclic object in the ($\infty$-)symmetric monoidal category $\mathcal{C}$, where $\Lambda$ is the cyclic ...
- 169
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
6
votes
1
answer
231
views
Weakly contractible $X$, but none of the maps $*\to X$ are cofibrations
Let $\mathrm{Top}$ be the category of all topological spaces and continuous maps. The Quillen model structure on $\mathrm{Top}$ has weak equvalences $W = \{ \text{weak homotopy equivalences} \}$, ...
- 331
13
votes
2
answers
1k
views
Categories on which one can determine all model structures?
Famously, there are exactly nine model structures on the category of sets, which are detailed here. In this case, one can exhaustively determine all six weak factorization systems and then see which ...
- 64k
14
votes
3
answers
656
views
Strøm model structures on the category of simplicial sets
Let $X,Y$ be simplicial sets. A simplicial homotopy is a simplicial map of the form $h:X\times\Delta^1\rightarrow Y$. There are two distinguished maps
$$
in_0:X\cong X\times\Delta^0\xrightarrow{1\...
- 5,596
0
votes
0
answers
85
views
Existence of covering space with trivial pullback map on $H^1$
I have seen somewhere the following claim (but can't remember where): let $M$ be a connected orientable closed smooth manifold with $b_1(M)=1$, then there exists a connected covering space $p:\tilde{M}...
3
votes
1
answer
243
views
Is every strongly causal spacetime purely electric?
Take a time 4-dimensional orinted Lorentzian manifold $(M,g)$.
A spacetime is called Strongly Causal at point $p$ if and only if for every neighbourhood $U$ of the point $p$ there exists a ...
- 612
3
votes
1
answer
144
views
Linearity of topological periodic cyclic homology
Let $A$ be an $E_\infty$ ring spectrum, $B$ a ring spectrum. Then if I understand correctly, $TP(A)$ is a ring spectrum by the lax monoidal property of $TP$. Suppose there is a map of ringed spectra ...
- 959
1
vote
0
answers
95
views
On the equivalence of two definitions of cohomological dimension for locally compact topological spaces
$\mathbf{The \ Problem \ is}:$ Let $X$ is a locally compact, separable metric space. Let $G$ be an abelian group. Now I came across two definitions of cohomological dimension of $X.$ One is the usual ...
1
vote
2
answers
269
views
Generalized cohomology on the one point space
I am reading Hatcher's algebraic topology for an assignment on generalized cohomology theories, and in section 4.E p. 447 he says the following
The wedge axiom implies that $h(\textit{point})$ is ...
- 11
1
vote
0
answers
144
views
Understanding the maps of the long exact sequence of cohomology from a Koszul complex
Suppose $\mathcal E$ is a bundle over a rational homogeneous smooth variety $X\subset \mathbb P^n$ such that the zero-locus $Z_s$ of a generic global section $s$ is a reduced zero-dimensional scheme. ...
- 111