All Questions
8,725 questions
7
votes
1
answer
468
views
An exact sequence involving THH
Disclaimer: I'm a relative beginner in this area. I'm trying to prove that if one has a commutative ring $R$ and a prime number $p$, then there is an exact sequence of the form
$$\DeclareMathOperator\...
- 73
98
votes
10
answers
14k
views
equivalence of Grothendieck-style versus Cech-style sheaf cohomology
Given a topological space $X$, we can define the sheaf cohomology of $X$ in
I. the Grothendieck style (as the right derived functor of the global sections functor $\Gamma(X,-)$)
or
II. the Čech ...
- 1,871
3
votes
1
answer
199
views
Subgroups of top cohomological dimension
Let $G$ be a geometrically finite group, i.e. there exists a finite CW complex of type $K(G,1)$.
By Serre's Theorem, every finite-index subgroup $H$ of $G$ satisfies $cd(H)=cd(G)$, but what about the ...
- 843
2
votes
1
answer
365
views
Correspondence between fundamental group and geometric properties of $X$
At the time of studing some algebraic topology I was wondering about the following.
Let $X$ be a topological space and $\pi_1(X)$ be its fundamental group.
If we assume some algebraic property of $\...
- 613
140
votes
7
answers
34k
views
Is the boundary $\partial S$ analogous to a derivative?
Without prethought, I mentioned in class once that the reason the symbol $\partial$
is used to represent the boundary operator in topology is
that its behavior is akin to a derivative.
But after ...
- 151k
8
votes
1
answer
369
views
Fiber product of spaces and cohomology
Let $Y \to X \leftarrow Z$ be a cospan of topological spaces and let $W = Y \times_X^h Z$ be their homotopy fiber product. I am interested in sufficient conditions on the map $Y \to X$ that ensure ...
- 9,853
4
votes
1
answer
282
views
Borel cohomology for circle actions on odd spheres
Suppose we have a $S^1$-action on the odd sphere $S^3$ as follows:
$$ \lambda \cdot (z_1, z_2) = (\lambda \cdot z_1, \lambda^2 . z_2)$$
I would like to understand the (Borel) equivariant cohomology of ...
- 169
6
votes
1
answer
380
views
The optimal ranges for the integral homological stability of $\operatorname{GL}_n(F)$'s for a field $F$
$\DeclareMathOperator\GL{GL}$
$\DeclareMathOperator\co{H}$
$\DeclareMathOperator\ko{K}$
$\DeclareMathOperator\trd{tr-deg}$
$\DeclareMathOperator{\ch}{char}$Given a field $F$ and a homological degree $...
- 1,836
2
votes
0
answers
164
views
Triviality of map $(\Sigma \theta)^*$
We know that there is a cofibration sequence
$$S^{4n+1}\xrightarrow{\theta}\Sigma^{4m-1} Q_{n-m} \rightarrow \Sigma^{4m-1} Q_{n-m+1} \rightarrow S^{4n+2}\xrightarrow{\Sigma\theta}\Sigma^{4m} Q_{n-m}.$$...
26
votes
1
answer
832
views
Are complex-oriented ring spectra determined by their formal group law?
To every complex-oriented ring spectrum $E$ there is associated a formal group law, which is a power series $F_E(x,y)\in E_*[[x,y]]$.
Suppose $E$ and $F$ are two complex-oriented ring spectra and ...
- 2,052
16
votes
0
answers
426
views
Is the oriented bordism ring generated by homogeneous spaces?
I am trying to find a Riemannian geometrically well-understood set of generators of the oriented bordism ring, including the torsion parts. By a set of generators, I mean that the set generates the ...
- 587
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
14
votes
2
answers
829
views
Can a cyclic group of prime order act on a rationally acyclic finite dimensional complex and have no fixed points?
Let $C_p$ b the cyclic group of order $p$, with $p$ a prime.
Is is possible for $C_p$ to act (cellularly) on a rationally acyclic finite dimensional CW complex $X$ with no fixed points?
Standard Smith ...
- 11.2k
1
vote
1
answer
192
views
Lie group framing and framed bordism
What is the definition of Lie group framing, in simple terms?
Is the Lie group framing of spheres a particular type of Lie group framing? (How special is the Lie group framing of spheres differed ...
- 447
4
votes
1
answer
421
views
4-manifold $M$ with intersection form of Leech lattice
Do we know any 4-manifold $M$ with intersection form of rank-24 Leech lattice?
Is $M$ smooth? Differentiable to which $n$-order?
Can $M$ be obtained by any surgery on 3 copies of E8 4-manifolds with ...
- 447
3
votes
0
answers
227
views
Classifying spaces beyond CW complexes
We know that for a reasonable topological group $G$ (say a compact Lie group) admits a classifying space for $G$-bundles within the category of countable CW complexes. That means, there is a space $BG$...
- 803
12
votes
0
answers
384
views
What are some examples of 3-dualizable $(\infty,2)$ categories?
From the cobordism hypothesis, we know that (the space of) symmetric monoidal functors from the $(\infty,3)$ category of framed cobordisms into a symmetric monoidal $(\infty,3)$ category is the same ...
- 2,356
5
votes
1
answer
346
views
Analogues of Sullivan Theory at a prime for coformality
In rational homotopy theory, one can study the rational homotopy and cohomology categories via an algebraic structure, respectively the rational Lie Algebra model and the Sullivan cdga model.
If I am ...
- 2,152
1
vote
1
answer
628
views
Cohomology of the amplitude space of unlabeled quantum networks
I am investigating a particular map from a product of three-spheres to the moduli space of (non-negative, real edge weight) networks. The map in question is
$$f: \smash{\left( \mathbb{S}^3 \right)}^N \...
- 571
9
votes
1
answer
322
views
Torsion and nilpotence of framed manifolds under the Pontryagin-Thom map
The framed Pontryagin Thom construction produces a graded ring isomorphism from the framed cobordism ring $\Omega^{fr}_*$ to the stable ring of homotopy groups of spheres $\pi_*(\mathbb{S})$. I have a ...
5
votes
1
answer
450
views
Can we prove that any number of algebraic data cannot be a complete invariant of a homotopy type?
The statement in the title seems to be generally accepted as true, but I have not seen proof. They are?
The strict formulation I have in mind is the following. By an algebraic category we mean the ...
- 6,962
3
votes
1
answer
238
views
1D topological defects in $d>3$ spatial dimensions
I am trying to construct a 1D topological defect solution in 4 spatial dimensions, i.e., a solution to some PDE (likely the equations of motion of some Lagrangian) on $\mathbb{R}^{4}$ which is ...
- 147
2
votes
1
answer
184
views
Lazard module structure of rings with formal elliptic curve
Recently in algebraic topology I was working with a certain graded ring $R$ equipped with an elliptic curve $C$. Now completion at the identity gives a 1-dimensional formal group $G$. This induces a ...
- 23
4
votes
0
answers
174
views
Do the nearby cycle and Beilinson's vanishing cycle functors commute?
Let $X$ be a complex algebraic variety with a pair of regular functions $f_1,f_2$. To these functions we can associate various functors: the nearby cycles functor $\Psi_{f_i}$, the vanishing cycles ...
- 1,071
66
votes
4
answers
6k
views
Is $\mathbb{R}^3 \setminus \mathbb{Q}^3$ simply connected?
Similarly is the complement of any countable set in $\mathbb R^3$ simply connected?
Reading around I found plenty of articles discussing the path connectedness $\mathbb R^2 \setminus \mathbb Q^2$ and ...
- 1,187
1
vote
0
answers
133
views
A question about fixed point set of the compact group actions
Let $G$ be an infinite compact Lie group acting on a compact space $X$.
Denote $F=F(G,X)=\{x\in X$ : $gx=x$ for all $g\in G\}$.
Show that if $H^*(B_{G_x};\mathbb{Q})=0$ for all $x \notin F$ and $T^1$ ...
- 1,367
27
votes
2
answers
798
views
Is there a flat manifold with trivial first homology?
Is there a closed flat manifold whose fundamental group has trivial abelianization?
The famous Hantzsche–Wendt flat manifold has fundamental group with finite abelianization.
- 29.1k
8
votes
0
answers
151
views
The James and Morse filtrations of homotopy groups
Denote by $JX$ the James construction on a path connected, well-pointed space $X$. This space is filtered by subspaces $X=J_1X\subseteq\dots\subseteq J_nX\subseteq\dots\subseteq JX$ and is the domain ...
- 5,596
22
votes
2
answers
978
views
Fixed-point free diffeomorphisms of surfaces fixing no homology classes
One of my graduate students asked me the following question, and I can't seem to answer it. Let $\Sigma_g$ denote a compact oriented genus $g$ surface. For which $g$ does there exist an orientation-...
- 313
2
votes
1
answer
100
views
Conjugacy problem in pure mapping class group of finitely-connected planar domain
Let $D$ be a finitely-connected planar domain, or, even more particularly, a domain obtained from the sphere $S^2$ by removing finitely many disjoint open topological disks. Let $\mathrm{PMCG}(D)$ be ...
- 41
7
votes
1
answer
179
views
Generalisation of Hirsch formula for the associativity of Steenrod's higher $\cup_2$ product with $\cup_1$ and cup products
For $f$, $g$ and $h$ cochains, the Hirsch formula is given as
$$ (f\cup g)\cup_1 h=f\cup (g\cup_1 h)+(-1)^{q(r-1)}(f\cup_1 h)\cup g.$$
Is there a more general formula that relates the associativity of ...
- 71
6
votes
1
answer
476
views
How to use a Heegaard diagram to retrieve the original 3-manifold that it represents?
(Disclaimer: I apologize that this is an introductory question for a forum like MathOverflow, but I have run out of ideas and resources to understand how this works, and I don't know where else to ask ...
- 377
1
vote
0
answers
151
views
Witten's QFT Jones polynomial work on Atiyah Patodi Singer theorem and $\hat A$ genus over Chern character
In Witten's 1989 QFT and Jones polynomial paper,
he wrote in eq.2.22 that
Atiyah Patodi Singer theorem says that the combination:
$$
\frac{1}{2} \eta_{grav} + \frac{1}{12}\frac{I(g)}{2 \pi}
$$
is a ...
- 447
7
votes
1
answer
490
views
Equivariant perverse sheaves and orbit stratification
Let $X$ be a complex algebraic variety with an action of a connected algebraic group $G$.
The forgetful functor from the category of $G$-equivariant perverse sheaves on $X$ to the category of perverse ...
- 3,446
2
votes
0
answers
179
views
Differentials in the Atiyah-Hirzebruch spectral sequence for a bounded generalized cohomology theory
$\newcommand{\res}{\mathrm{res}}$Let $X$ be a connected finite CW complex, and let $E$ be a bounded spectrum. For simplicity, let me assume that it has homotopy groups concentrated in degrees 0,1,2.
...
- 1,001
5
votes
1
answer
195
views
Alternate way to group complete a homotopy commutative topological monoid
Let $M$ be a topological monoid that is homotopy commutative. I've been trying to understand the proof of the group completion theorem from Hatcher's notes. Roughly speaking, this theorem says that ...
- 51
40
votes
2
answers
2k
views
Can the nth projective space be covered by n charts?
That is, is there an open cover of $\mathbb{R}P^n$ by $n$ sets homeomorphic to $\mathbb{R}^n$?
I came up with this question a few years ago and I´ve thought about it from time to time, but I haven´t ...
- 10.6k
6
votes
2
answers
437
views
Presentation of the fundamental group of a complex variety
Let $X$ be a connected smooth complex algebraic variety and $Z=\bigcup_{i=1}^r Z_i$ be a union of smooth connected hypersurfaces, satisfying that each two intersect transversally. Assume for ...
- 541
2
votes
0
answers
175
views
Minimal first Pontryagin class $p_1=1$?
From Hirzbuch theorem,
the signature of 4-manifold $\sigma = p_1/3$ with the first Pontryagin class $p_1$.
I know that the $\sigma=p_1/3 =1$ so $p_1=3$ for complex projective space $\mathbb{P}^2$.
Is ...
- 447
23
votes
1
answer
1k
views
A property of even continuous functions on the sphere
This question is inspired by On moments of inertia of planar and 3D convex bodies.
Let $f:{\mathbb R}^3\setminus\{0\}\to{\mathbb R}$ be an even homogeneous ($f(kx)=f(x)$ for all real $k\neq 0$) ...
- 91.8k
26
votes
1
answer
1k
views
Spheres with the same homotopy groups
What is known about the existence of other pairs of spheres (such as $S^2$ and $S^3$) whose homotopy groups coincide starting from some index.
A sufficient condition for this is the existence of a ...
- 6,962
3
votes
1
answer
228
views
Does $H^3\times I$ admit a Kähler metric?
Let $H^3$ be the Heisenberg manifold. It is known that the first betti number of $H^3\times S^1$ is odd and therefore it does not support any Kähler metric. Now let $I=(0,1)$ or $I=[0,1]$, does it ...
- 661
3
votes
2
answers
677
views
Does the cohomology Bockstein homomorphism map to the homology Bockstein homomorphism under Poincarè duality?
Given a manifold $X$ and short exact sequence of abelian groups
$$
1\rightarrow A_1\overset{\iota}{\rightarrow} A_2\overset{\pi}{\rightarrow} A_3\rightarrow 1
$$
we get the Bockstein map in cohomology ...
- 813
54
votes
10
answers
12k
views
Intuition behind Thom class
The Thom class and Thom isomorphism theorem for oriented vector bundles are proven ( at least to my knowledge) by induction on the open covers and some manipulation with Mayer-Vietoris sequences.
...
- 1,357
7
votes
1
answer
313
views
From the *usual* nerve of topological categories to $\infty$-categories
It is standard from work of Joyal and Lurie that there is a Quillen equivalence between the model category of simplicially enriched categories $Cat_\Delta$ and $\mathcal{S}\text{et}_\Delta$ with the ...
- 367
7
votes
1
answer
442
views
Is anything known about de Rham $K(\pi,1)$'s?
Let $X$ be a connected qcqs scheme. We say that $X$ is a (étale) $K(\pi,1)$ if for every locally constant constructible abelian sheaf $\mathscr{F}$ on $X$ and every geometric point $\overline{x}$ the ...
- 771
31
votes
7
answers
3k
views
Why are we interested in permutahedra, associahedra, cyclohedra, ...?
The following families of polytopes have received a lot of attention:
permutahedra,
associahedra,
cyclohedra,
...
My question is simple: Why?
As I understand, at least the latter two were ...
- 13.6k
3
votes
1
answer
192
views
Rational group homology of an infinite product of finite groups
Let $G_{1}, G_{2}, \cdots$ be a countably infinite sequence of finite groups. It is well-known that the group homology $H_{n}(BG_{i};\mathbb{Q})=0$ for any $n\geq 1$.
Let $X=\prod^{\infty}_{i=1}BG_{i}$...
- 1,069
1
vote
2
answers
172
views
Reference for choosing a path lifting function?
I recall having seen discussion of a Hurewicz or Serre fibration
equipped with a chosen path lifting function. Citation??
- 301
25
votes
1
answer
1k
views
Homotopy type of Diff(ℝP³)
$\DeclareMathOperator\Diff{Diff}$I am looking for the paper computing the homotopy type of the group $\Diff(\mathbb{R}P^3)$ of diffeomorphisms of the $3$-dimensional real projective space $\mathbb{R}P^...
