All Questions
8,725 questions
4
votes
1
answer
282
views
Borel cohomology for circle actions on odd spheres
Suppose we have a $S^1$-action on the odd sphere $S^3$ as follows:
$$ \lambda \cdot (z_1, z_2) = (\lambda \cdot z_1, \lambda^2 . z_2)$$
I would like to understand the (Borel) equivariant cohomology of ...
- 169
2
votes
0
answers
84
views
Euler class of extension of free nilpotent groups
Fix some $n \geq 2$. For $k \geq 1$, let $N_k$ be the free $k$-step nilpotent group on $n$ generators, i.e., the quotient of the free group $F_n$ by the $(k+1)^{\text{st}}$ term $\gamma_{k+1}(F_n)$ ...
- 21
1
vote
1
answer
203
views
Moving of sphere embedding and its interior defined by Jordan-Brouwer separation theorem
Let $f_1:\mathbb S^{n-1}\rightarrow \mathbb R^n$ be a continuous embedding, where $\mathbb S^{n-1}$ is the unit sphere of dimension $n-1$, and a point $x$ in the interior of $f_1(\mathbb S^{n-1})$ ...
- 435
8
votes
1
answer
369
views
Fiber product of spaces and cohomology
Let $Y \to X \leftarrow Z$ be a cospan of topological spaces and let $W = Y \times_X^h Z$ be their homotopy fiber product. I am interested in sufficient conditions on the map $Y \to X$ that ensure ...
- 9,853
2
votes
1
answer
135
views
Compact objects in persistence modules and interval decomposition
$\newcommand\Mod{\mathrm{Mod}}\DeclareMathOperator\Fun{Fun}$If $k$ is a field, a persistent $k$-module is a functor $\mathbb{R}\to \Mod_k$ where $\mathbb{R}$ is a poset under the natural ordering of $\...
- 213
9
votes
1
answer
444
views
Hochschild cohomology of a group algebra
Let $K$ be a field and $G=\pi_1(\Sigma_g)$ the surface group of genus $\geq 2$. I want to know the Hochschild cohomology of the group algebra $A=K[G]$ with coefficients in $A$ and $A\otimes A$, namely,...
- 985
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
1
vote
0
answers
90
views
Cohomological dimension of the kernel of a homomorphism induced by a singular fibration
I have a very concrete question. Let $N={\Bbb C}-\{\pm 1\}$, and ${\Bbb Z}_2$ is acting on $N$ by rotation around $0$. Consider $M=\{(x,y)\in N^2\ |\ {\Bbb Z}_2x\neq {\Bbb Z}_2y\}$. Let $p:M\to N$ be ...
- 585
6
votes
1
answer
203
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
13
votes
2
answers
538
views
How many automorphisms are there of the category of filtered spectra?
Dold-Kan type theorems tell us that lots of categories are Morita-equivalent to the simplex category $\Delta$. In other words, there are a lot of stable $\infty$-categories which are secretly ...
- 64k
5
votes
0
answers
166
views
Equivalent descriptions of equivariant K-theory
I am looking at references for computing $$K_{T}(G/H)$$ where $G$ is a compact connected Lie group with maximal torus $T$, and $H\subset G$ is a corank one Lie subgroup such that $G/H\cong S^{2k-1}$ ...
- 51
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
1
vote
1
answer
152
views
Compact locus in (ordered) configuration spaces
Let $\mathit{Conf}_n(\mathbb{R}^2)$ be the configuration space of $n$ ordered distinct points in the plane. I'd like to know if the topological subspace $C_n$ consisting of points $(p_1,...,p_n)$ with ...
- 1,125
2
votes
0
answers
192
views
Completeness of the Infinity Category of A-infinity Categories
Is the infinity category of $A_{\infty}$-categories complete? By complete I mean do there exist arbitrary homotopy limits in the infinity category of $A_{\infty}$-categories?
I felt like this result ...
- 341
1
vote
0
answers
155
views
Lifting action of torus to torus bundle
Preamble: Let $X$ be a simply connected smooth manifold and $P \to X$ be a principal $T^\ell$ bundle on it.
Let $\phi$ be a smooth action of $T^k$ on $X$.
The paper "Lifting compact group actions ...
- 205
8
votes
1
answer
779
views
Connеcted components of irreducible algebraic varieties
I am wondering what is the possible (or maximum) number of connected components for an irreducible algebraic variety in $\mathbb R^n$ defined by a degree $d$ polynomial (i.e. hypersurface) in $\mathbb ...
- 185
5
votes
0
answers
249
views
Aspherical space whose fundamental group is subgroup of the Euclidean isometry group
Let $M$ be a smooth, compact manifold without a boundary, with its universal covering $\tilde{M} = \mathbb{R}^n$. If there exists an injective homomorphism $h: \pi_1(M) \rightarrow O(k) \ltimes \...
- 661
5
votes
1
answer
328
views
Is the inclusion of the maximal torus in a simply connected compact Lie group null-homotopic?
Let $G$ be a simply connected compact Lie group and $T$ its maximal torus with inclusion $i:T \hookrightarrow G$.
By simply connectedness of the group $G$ and asphericity of the torus $T$, the induced ...
- 53
3
votes
2
answers
285
views
Cut a homotopy in two via a "frontier"
Consider a space $G$ obtained by glueing two disjoint cobordisms (the fact that they are might be irrelevant, assume they are topological spaces at first) $L$ and $R$ on a common boundary $C$.
(...
3
votes
1
answer
148
views
Homotopy groups of $K(n)$-local $E_n$-modules are $L$-complete
Let $E_n$ be the $n$-th Morava $E$-theory and let $K(n)$ denote the $n$-th Morava $K$-theory.
Question: If $M$ is a $K(n)$-local $E_n$-module, then are the homotopy groups $\pi_*(M)$ $L$-complete? (...
- 177
9
votes
0
answers
147
views
degree 1 maps for bordism homology
Let $f\colon X \to Y$ be a degree 1 map between closed oriented manifolds. Then the induced homomorphism between the homology groups is surjective up to torsion.
Can one say something similar about (...
2
votes
1
answer
284
views
Is there an $X$ and a non-surjective continuous function $f:X\to X$ such that $f\simeq {\rm id}_X$ but $f(X)$ and $X$ are not homotopy equivalent?
Let $X\subset \mathbb{Z}^n$ be a finite set (vertex set) and $1\leq u\leq n$. For $x=(x_1 ,\ldots ,x_n)\neq y=(y_1 ,\ldots ,y_n) \in X$, we define the adjacency $\kappa_u$ on $X$ as follows: $x\sim_{\...
- 1,182
2
votes
0
answers
370
views
What is the nerve of this category?
If $\mathcal{C}$ is a thin category, we call $U \subseteq \mathrm{Ob}(\mathcal{C})$ open if for every object $X \in U$ and any morphism $X \to Y$, we also have $Y \in U$. This declares an Alexandrov ...
- 612
5
votes
0
answers
150
views
Analytical Dold-Thom
Let's $X$ be a projective smooth variety over a field that has an embedding into $\mathbb{C}$. Let's denote the infinite symmetric power of $X$ by $\text{Sym}^{\infty}(X)$. Denote the algebraic ...
- 5,901
2
votes
2
answers
276
views
The complex $K$-theory of the Thom spectrum $MU$
The Atiyah-Hirzebruch spectral sequence is a strong computational tool that yields several interesting computation in (co)homology. I want to know whether $K_\ast(MU)$ and $K^\ast(MU)$ have been ...
- 21
2
votes
1
answer
365
views
Correspondence between fundamental group and geometric properties of $X$
At the time of studing some algebraic topology I was wondering about the following.
Let $X$ be a topological space and $\pi_1(X)$ be its fundamental group.
If we assume some algebraic property of $\...
- 613
1
vote
0
answers
161
views
Higher dimensional Seifert surfaces and link numbers of higher knots
In 3-manifold topology, the notion of Seifert surface is well known. It is then used to define link numbers of knots.
Question: Consider embedding $N^n \rightarrow M^{2n+1}$ of n-dimensional manifold $...
- 593
11
votes
2
answers
858
views
Spectral sequences and short exact sequences
Suppose I take a short exact sequence of filtered chain complexes:
$$0\to A\xrightarrow{p} B\xrightarrow{q} C\to 0$$
We assume that $p$ and $q$ are filtration-preserving, so that $p(F_rA)\subseteq ...
- 721
2
votes
0
answers
55
views
Tangential normal invariant isomorphism
Recently, I was reading the paper "Finite Group Actions on Kervaire Manifold" by Crowley, Hambolton. But I am having problem understanding a definition. Here it is,
In page 15-16 they are ...
2
votes
0
answers
354
views
Square-zero extensions mod $p^n$
$\DeclareMathOperator\LMod{LMod}\DeclareMathOperator\Mod{Mod}\DeclareMathOperator\Sp{Sp}$A square-zero extensions of rings is, conceptually, a map of rings $R \to A$ such that any two elements in the ...
- 68
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
3
votes
0
answers
125
views
Does symmetric product functor preserve fibrations?
I know that the symmetric product is a functor, cf: https://en.wikipedia.org/wiki/Symmetric_product_(topology)#Functioriality.
My question is, does it preserve fibrations in the category of ...
- 506
5
votes
0
answers
133
views
Division of fibration by $\Sigma_{n}$ gives Serre fibration
This is related to a question posted on StackExchange: https://math.stackexchange.com/questions/4776877/left-divisor-of-a-fibration-by-compact-lie-group-is-a-fibration. The question there had received ...
- 311
3
votes
1
answer
402
views
The derived category of $p$-complete abelian groups is comonadic over the derived category of $\mathbb F_p$-vector spaces?
$\DeclareMathOperator\Mod{Mod}\DeclareMathOperator\Ext{Ext}$Let $p$ be a prime. The adjunction
$$\mathbb F_p \otimes_\mathbb{Z} (-) : \Mod(\mathbb Z) \rightleftarrows \Mod(\mathbb F_p) : U $$
descends ...
- 64k
4
votes
1
answer
489
views
Cohomology of finite symmetric products of manifolds
Let $M$ be a closed, orientable manifold of dimension $k$. I am looking for results to determine explicitly the (co)homology groups and/or cohomology ring structure (in integer or rational ...
- 506
1
vote
0
answers
64
views
Physical measure of a dynamical system in terms of its density
Let $f$ be a $\mathcal{C}^1$ vector field on a compact subset $M \subset \mathbb{R}^n$. We define a dynamical system by
$$\dot{x}(t)=f(x(t))$$
In ergodic theory, the occupation measure is
$$\mu_{x, T}(...
- 247
1
vote
0
answers
131
views
Can we construct a general counterexample to support the weak whitney embedding theorm?
The weak Whitney embedding theorem states that any continuous function from an $n$-dimensional manifold to an $m$-dimensional manifold may be approximated by a smooth embedding provided $m > 2n$.
...
- 109
2
votes
0
answers
147
views
Explicit S-duality map
$\DeclareMathOperator{\Th}{Th}$
The Thom space of a closed manifold $M$ ($\Th(M)$) is the $S$-dual to $M_+ (=M \cup \{pt\})$. Let $M$ be embedded in $\mathbf R^{n+k}$. I found the duality map from $S^...
4
votes
2
answers
673
views
Determine monodromy representation from local system
Let $X$ be a path-connected manifold nice enough such it's universal covering
space $p:\widetilde{X} \to X$ exists, $k$ a field. Then there exist a wellknown
correspondence
$$
\{\textit{linear}\text{ ...
- 619
13
votes
1
answer
387
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
8
votes
1
answer
205
views
Connectivity of the space of transverse vector fields
Suppose we have a smooth, closed manifold $M$ of dimension $n$ and connectivity $k$. What can we say about the connectivity of the space of all tangent vector fields on $M$ that are transverse to the ...
- 2,062
0
votes
0
answers
162
views
Gluing faces of n-cube
Assuming $C_n$ be the $n$-cube, the intersection of $C_n$ with a supporting hyperplane $H(P, v)$ is called a face or more precisely a $d$-face if the dimension is $d$.
Let $f_0$ and $f_1$ be faces ...
- 53
34
votes
6
answers
4k
views
Why study finite topological spaces?
In rereading Thurston's essay On Proof and Progress in Mathematics I ran across this passage:
… this means that some concepts that I use freely and naturally in
my personal thinking are foreign to ...
- 737
3
votes
0
answers
70
views
Is every finite spectrum $X$ $K(h)$-locally equivalent to a finite spectrum $Y$ with $\dim (K(h)_\ast Y) = \dim ((H\mathbb F_p)_\ast Y)$?
Let $X$ be a finite spectrum and $K = K(h)$ be the $h$th Morava $K$-theory at the prime $p$. Then $\dim_{K_\ast} K_\ast X$ is increasing in $h$, and eventually constant at $\dim H_\ast(X,\mathbb F_p)$....
- 64k
1
vote
1
answer
149
views
Relative $G$-equivariant homology groups
Let $X$ be a free $G$-CW-complex with $G$-equivariant cell filtration by
$n$-skeleta $X_0 \subset \dots \subset X_n \subset \dots \subset X$ (for
rigorous definition see
Chap. II, p. 98 in linked ...
- 6,048
5
votes
0
answers
106
views
Classifying spaces of crossed modules
Let $\mathcal{G}$ be a strict topological $2$-group, i.e. a strict $2$-category with a single object, a space of invertible $1$-morphisms, a space of invertible $2$-morphisms and continuous structure ...
- 7,637
11
votes
1
answer
381
views
Chromatic representation theory of the symmetric groups?
We know that in characteristic 0, the group ring of the symmetric group $\Sigma_n$ splits via one idempotent for each partition of $n$.
In characteristic $p$, I believe the analogous statement is that ...
- 64k
1
vote
0
answers
226
views
Cohomology in families of normal varieties
Let $f : X \to Y$ be a flat proper morphism of complex varieties whose fibers are normal varieties. Is it true that $\mathrm{dim}_{\mathbb{Q}} H^i(X_t, \mathbb{Q})$ is constant?
For non-normal fibers, ...
- 3,720
2
votes
0
answers
435
views
Classification of homotopy types of topological spaces
Do higher groups classify the homotopy types of topological spaces?
We may assume $\pi_n$ of the topological spaces are all finite
and $\pi_n =0$ for large enough $n$.
For example, if only $\pi_1 \neq ...
- 4,826
9
votes
2
answers
421
views
How do these definitions of factorization algebra compare?
Question
Several sources define (homotopy) factorization algebras in a seemingly
different manner (I am looking at [CG], [Gi], and
[CFM].) I wish to know how they compare with each other.
I apologize ...
- 2,292