All Questions
9,056 questions
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
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
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
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
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
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
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
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
441
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 ...
- 721
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
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
2
votes
1
answer
271
views
Apropos of two groups being globally isomorphic iff they are isomorphic
Denote by $\mathcal P(S)$ the semigroup obtained by endowing the non-empty subsets of a "ground semigroup" $S$ (written multiplicatively) with the operation of setwise multiplication induced ...
- 10.5k
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^...
19
votes
3
answers
1k
views
Are Chern classes well defined up to contractible choice?
The Chern classes are, by definition, cohomology classes. And
cocycle representatives of the Chern classes are not unique.
But it might be the case that cocycle representatives of the Chern classes ...
- 43.2k
3
votes
1
answer
358
views
A question about Bockstein homomorphisms
For $r\geq 1$, we have the following short exact sequence
$$0\rightarrow \mathbb{Z}/2^r\mathbb{Z} \xrightarrow{\cdot 2^r}\mathbb{Z}/2^{2r}\mathbb{Z} \xrightarrow{\bmod2^r}\mathbb{Z}/2^r\mathbb{Z}\...
- 545
5
votes
2
answers
375
views
Monomorphisms of diagrams in an $\infty$-category
Let $f,g\colon K\to \mathcal{C}$ be diagrams in a nice $\infty$-category $\mathcal{C}$. I have two general questions:
If I have a natural transformation $\eta\colon f\Rightarrow g$ which is a ...
- 10.4k
6
votes
0
answers
209
views
"Inclusion" between higher categories of framed bordisms?
Let $\mathrm{Bord}_n$ be the bordism $(\infty, n)$-category of unoriented manifolds.
It can be viewed as an $(\infty, n+1)$-category whose $n+1$-morphisms are equivalences.
If $n$ is large enough, ...
- 890
5
votes
2
answers
321
views
Reedy fibrancy and composition in Segal spaces
I am going through V. Hinich's "Lectures on Infinity Categories" and I have a (possibly trivial) question on Segal spaces.
We define Segal space to be a bisimplicial set $X$ which is fibrant ...
- 1,759
82
votes
12
answers
15k
views
Compelling evidence that two basepoints are better than one
This question is inspired by an answer of Tim Porter.
Ronnie Brown pioneered a framework for homotopy theory in which one may consider multiple basepoints. These ideas are accessibly presented in his ...
- 22.1k
2
votes
1
answer
284
views
Is there an $X$ and a non-surjective continuous function $f:X\to X$ such that $f\simeq {\rm id}_X$ but $f(X)$ and $X$ are not homotopy equivalent?
Let $X\subset \mathbb{Z}^n$ be a finite set (vertex set) and $1\leq u\leq n$. For $x=(x_1 ,\ldots ,x_n)\neq y=(y_1 ,\ldots ,y_n) \in X$, we define the adjacency $\kappa_u$ on $X$ as follows: $x\sim_{\...
- 1,182
1
vote
0
answers
145
views
Determine the Eilenberg-MacLane spaces on the right-handed side of this Whitehead tower?
What and how to determine the Eilenberg-MacLane spaces on the right-handed side of this Whitehead tower?
Namely, how do we know
$$
K(Z_2,1)?, \quad K(Z_2,2)?, \quad K(Z,4)?
$$
Naively -- in each step ...
- 447
9
votes
2
answers
443
views
Simplicial sets with horn filling conditions up to some fixed degree
Let $X_\bullet$ be a simplicial set such that some horn filling condition (inner horns fill/inner horns fill uniquely/all horns fill) holds up to dimension $n$ (i.e. for $\Lambda_i[p]$ for all $p\leq ...
- 1,109
1
vote
0
answers
240
views
Examples of when $X$ is homotopy equivalent to $X\times X$
I was thinking about this question the other day: When is a topological space $X$ homotopy equivalent to $X\times X$ (with the product topology)? This is essentially a cross-post of this MSE question.....
7
votes
0
answers
254
views
$C^0$-limit of volume-preserving maps on $\mathbb R^n$
Let $f_k:B_1\rightarrow \mathbb R^n$ be a sequence of injective differentiable volume-preserving maps (i.e. $\mu(f_k(A))=\mu(A)$ for any measurable $A\subset B_1$) that converges uniformly to $f:B_1\...
- 435
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
30
votes
1
answer
1k
views
Are homeomorphic representations isomorphic?
Let $G$ be a finite group. Let $V_1, V_2$ be two finite-dimensional real representations. Suppose $f: V_1 \to V_2$ is a $G$-equivariant homeomorphism. Can one conclude that $V_1$ and $V_2$ are ...
- 803
4
votes
0
answers
116
views
Finding inverses of certain elements in the set of normal invariants of a smooth manifold
Let, $V$ denote the Stiefel manifold of 2-frames $V_{10,2}$ . Consider the the map $S_\text{diff} (V) \xrightarrow{\eta} N_\text{diff} (V) $ in the surgery exact sequence of a smooth manifold. . ...
- 141
5
votes
1
answer
632
views
Is the singular simplicial complex functor $\operatorname{Sing}_\bullet:\operatorname{Top} \to \operatorname{sSets}$ fully faithful for nice spaces?
$\DeclareMathOperator{\Hom}{Hom} \DeclareMathOperator{\Sing}{Sing} \DeclareMathOperator{\Top}{Top}
\DeclareMathOperator\sSets{sSets}$Is there a category of “sufficiently nice” topological spaces such ...
- 513
15
votes
4
answers
2k
views
Cohomology ring of mapping torus
A mapping torus, $M \rtimes_\varphi
S^1$, is a fiber bundle over $S^1$ with fiber $M$, where $\varphi$ is an element of mapping class group of $M$, describing the twist around $S^1$.
For $M=S^1\times ...
- 4,826
16
votes
1
answer
776
views
The second stable homotopy group
I am interested in the low-degree stable homotopy group $\pi_2^{s}(X)$ of a path-connected space $X$. Using the Atiyah-Hirzebruch spectral sequence, we have the short exact sequence $0\to H_1(X,\...
- 663
6
votes
3
answers
393
views
Structure theorem for a class of idempotent monoids (where $xy = x$ or $xy = y$ for all $x, y$)
Question. Is there any structure theorem for the class of monoids $H$ with the property that $xy = x$ or $xy = y$ for all $x, y \in H$? Or does this look hopeless for some good reasons?
A monoid with ...
- 10.5k
2
votes
0
answers
169
views
How a circle $S^1$ acts on the Cayley plane $OP^2$ with exactly three fixed points?
The complex projective $CP^2$, the quaternionic projective space $HP^2$, and the octonionic projective space $OP^2$ each admit a circle action with $3$ fixed
points.
The circle action on $HP^2$ can be ...
- 103
2
votes
0
answers
137
views
Tensor product of objectwise weak homotopy equivalences of $\mathcal{M}$-spaces
I consider the enriched category $[\mathcal{M}^{op},\mathrm{Top}]$ of enriched functors (I call them $\mathcal{M}$-spaces) from the enriched small category $\mathcal{M}^{op}$ to the enriched category $...
- 3,901
1
vote
1
answer
108
views
Simplicial cochain representing the pullback of a class Poincaré dual of a submanifold
Let $K$ be a simplicial complex of dimension $n$, $M$ be a topological manifold, and $f \colon |K|
\to M$ be a continuous map. Let $X$ be an embedded manifold in $M$ of codimension $n$, such that
$f(|...
- 111
1
vote
0
answers
118
views
A question about cohomology with local coefficient
Let's consider the next theorem.
Theorem
[The cohomology Leray-Serre Spectral sequence] Let $R$ be a
commutative ring with unit. Given a fibration $F\hookrightarrow E\overset{p}{%
\rightarrow }B$, ...
- 1,367
8
votes
0
answers
226
views
A variation of necklace splitting
Our problem is the following:
Let $n$ and $k$ be integers.
We are given two (unclasped) necklaces, each with $n$ colored stones: a top necklace which has $k$ colors and a bottom necklace which has 2 ...
- 81
6
votes
1
answer
350
views
Reference for a property of Dehn twists
I was reading The symplectic Floer homology of a Dehn twist by P. Seidel, which you can find here.
In Lemma 3(ii) the following topological property of Dehn twists is stated without proof:
Let $\...
- 293
6
votes
1
answer
407
views
Is there a clear pattern for the degree $2n$ cohomology group of the $n$'th Eilenberg-MacLane space?
Let $G$ be a finite abelian group, and its higher classifying space is $B^nG=K(G,n)$. For $n=1$ it is well known that $H^2(B G, \mathbb{R}/\mathbb{Z}) \cong H^2(G,\mathbb{R}/\mathbb{Z})$ is isomorphic ...
- 813
4
votes
1
answer
419
views
Faithful locally free circle actions on a torus must be free?
Is it true that every faithful and locally smooth action $S^1 \curvearrowright T^n$ is free?
I know such an action must induce an injection $\rho:\pi_1(S^1)\to\pi_1(T^n)$.
Another related question is: ...
- 385
13
votes
2
answers
927
views
When did the Joyal model structure on simplicial sets originate?
Some of the earliest writings on the Joyal model structure on simplicial sets include Jacob Lurie's account in Higher Topos Theory from 2006,
as well as Joyal's own account in The Theory of Quasi-...
- 37.8k
3
votes
2
answers
253
views
On algebraic topology of coset complexes without geometry
I'm interested in understanding the algebraic topology of "coset complexes" from a "combinatorial" perspective (i.e., without relying on geometric realizations of the complexes). ...
- 173
4
votes
1
answer
424
views
Homotopy groups of mapping cylinder
Let $f:K\to X$ be a map, with mapping cyliner $M=X\cup_{f}(K\times \{ 1\})$. 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 $\alpha :S^{...
- 287
3
votes
1
answer
388
views
When does a fibre bundle induce a long exact sequence in homotopy groups of mapping spaces?
Given a smooth fibre bundle of (compact, say) manifolds $F \to E \to M$, with $\pi : E \to M$ the projection, and $i_p : F \to E$ the inclusion of $F$ into any fibre $\pi^{-1}(p)$, we get, for any ...
- 1,763
2
votes
1
answer
235
views
What is the homotopy type of the smash power of Moore spectra $(S/2)^{\otimes n}$?
Let $S/2$ be the mod $2$ Moore spectrum, and let $n \in \mathbb N$.
Question:
What is the homotopy type of the $n$th smash power $(S/2)^{\otimes n}$?
Notes:
When $p$ is odd, we have $S/p \otimes S/p =...
- 64k
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
5
votes
0
answers
188
views
Are there known minimal models for the cohomology of semisimple Lie algebras?
My student and I recently found a cute construction of a minimal model for the cohomology of a Lie algebra $\mathfrak{g}$. This is a "minimal model" in the sense that it is a minimal chain-...
- 1,335
5
votes
1
answer
191
views
Regular polyhedral spaces
By symmetrically gluing together opposite faces of a dodecahedron together, one of three spaces can be obtained, depending on the angle the faces are rotated by before twisting. In fact, this can be ...
- 2,773
9
votes
1
answer
339
views
Stably-framed cobordism $(\infty,n)$-category
In Lurie's treatment of the cobordism hypothesis, the domain is $\mathsf{Bord}^{fr}_n$, the symmetric monoidal $(\infty,n)$-category of $k$-bordisms with $n$-framing for $0\leq k\leq n$.
If I ...
- 663