All Questions
9,056 questions
3
votes
0
answers
83
views
Cohomological interpretation of gluing conditions
This question comes from a problem of theoretical physics. Stated in its simplest form, there is a complex line bundle over $S^1$. For each $z \in S^1$, the fiber $F_z$ is the complex eigenspace of a ...
- 141
2
votes
0
answers
146
views
Fundamental groups between surjective proper fibrations
Let $f:X\to Y$ be a surjective proper morphism between connected (but might be reducible) complex quasi-projective varieties. Assume that each fiber of $f$ is connected. Can we conclude that ${\rm Im}(...
- 527
6
votes
1
answer
487
views
Topology change induced by small perturbation
Consider the surface $S_{\epsilon}$ defined as:
\begin{align}
%S&=\{\vec x \in \mathbf{R}^3: x=0\}, \\
S_{\epsilon}&=\{\vec x \in \mathbf{R}^3:f_{\epsilon}(\vec x)\equiv\epsilon ((x^2 + y^2 - ...
- 521
12
votes
1
answer
624
views
Stone–Čech compactification as a semigroup
Let $G$ be a topological group (we can assume that $G$ is countable and discrete) and let $\beta(G)$ be the Stone–Čech compactification of $G$. It is known that $\beta(G)$ can be turned into a left ...
5
votes
0
answers
227
views
Higher coherences in $A_\infty$-spaces
Recently I was learning about $A_\infty$-spaces. The slogan what they are is: "Homotopy associative H-spaces with higher coherences for the associativity".
When I first heard the slogan, I ...
- 2,998
12
votes
2
answers
1k
views
Algebraic topology and homotopy theory with simplicial sets instead of topological spaces
To quote Kerodon:
In fact, it is possible to develop the theory of algebraic topology in entirely combinatorial terms, using simplicial sets as surrogates for topological spaces.
A similar quote can ...
- 123
7
votes
0
answers
185
views
How does nullification of $K(\mathbb{Z}, 2)$ compare to 1-truncation?
Let $B$ be a type (or space). A type (or space) is $B$-null if the canonical map $X \to X^B$ is an equivalence. The $B$-nullification of an arbitrary type $X$ is a $B$-null type $\bigcirc_B X$ ...
- 4,378
3
votes
0
answers
249
views
Grothendieck schemes and the Sheffer differential op calculus (Rota, Roman, et al. finite operator calculus)
In "Left differential operators on non-commutative algebras" on p. 4, Michiel Hazewinkel displays "precisely the right definition of differential operator" as
$$D\; X^n = F(\tfrac{...
- 10.5k
3
votes
1
answer
275
views
Is $\Sigma^\infty_+ O(n)^\vee$, the Spanier-Whitehead dual of the orthogonal group, an $A_\infty$-ring spectrum?
Recently, Ching and Salvatore have proven that the $E_n$ operad is Koszul self dual. While thinking about the analogous question for the framed $E_n$ operad, I realized there is an obvious first ...
- 5,849
4
votes
0
answers
176
views
Is there any results about the stable (or unstable) cohomology operations on cohomology of Lie groups?
$\DeclareMathOperator\SU{SU}$For the $\mod p$ singular cohomology of classical Lie groups, such as $H^*(\SU(n); \mathbb{Z}/p\mathbb{Z})$, there are well known results about the actions of the stable ...
- 71
20
votes
3
answers
715
views
Which elements of $H^2(M,\mathbb{Z}/2)$ are the $w_2(E)$ for a real bundle $E$?
Any element of $H^1(M,\mathbb{Z}/2)$ is the $w_1(E)$ of a real line bundle $E$ over $M$.
I wonder how to characterize (probably using the Steenrod squares) which elements of $H^2(M,\mathbb{Z}/2)$ are ...
- 6,133
93
votes
9
answers
37k
views
Is Mac Lane still the best place to learn category theory?
For a student embarking on a study of algebraic topology, requiring a knowledge of basic category theory, with a long-term view toward higher/stable/derived category theory, ...
Is Mac Lane still ...
2
votes
0
answers
118
views
Fibering a certain hyperplane arrangement complement over a fiber-type arrangement complement
For motivation I recall two classical examples of locally trivial fiber bundle projections, due to Brieskorn (in `Sur les groupes de tresses').
${\Bbb C}^n-\cup_{i\neq j}\{(z_1,z_2,\ldots ,z_n): z_i=\...
- 585
21
votes
2
answers
1k
views
Does Waldhausen K-theory detect homotopy type?
Recall that $A(X)$, the K-theory of a connected, pointed space X, is defined as the K-theory spectrum of the ring spectrum $\Sigma^\infty_+ \Omega X$ (or via a plethora of alternative definitions). Is ...
- 5,849
2
votes
1
answer
244
views
Explicit 3-cocycle of group cohomology of dihedral group and generalization to other semidirect products
The dihedral group $D_8$ can be presented as $(\mathbb{Z}_2\times \mathbb{Z}_2)\rtimes _{\rho}\mathbb{Z}_2$, where the last factor acts on $\mathbb{Z}_2\times \mathbb{Z}_2$ as
$$
\rho_1(a,b)=(b,a) \ .
...
- 813
9
votes
2
answers
1k
views
Hodge dual of de Rham cohomology and singular cohomology
We know that the de Rham cohomology is isomorphic to the singular cohomology, does the Hodge dual of differential forms induce a dual operation on de Rham cohomology, hence also on singular cohomology?...
- 10.5k
2
votes
0
answers
85
views
Turning cocycles in cobordism into an inclusion or a fibering
By the classical Pontryagin-Thom construction, we know that the cobordism group $\Omega^n_U(X)$ is represented by cocyles
$$ M\hookrightarrow X\times \mathbb{R}^{2k}\rightarrow X,$$
where $M$ is a ...
- 171
1
vote
1
answer
279
views
Ways to prove that $n$-component Brunnian link is nontrivial
The attached image shows a way to construct an $n$-component Brunnian link for any $n\geq 3$. That is, this link is not trivial, but deleting any of its components makes the new link trivial. The ...
- 214
5
votes
1
answer
223
views
Rational G-spectrum and geometric fixed points
For a finite group $G$, how is a rational $G$-spectrum $X$ detected by the geometric fixed point functor $\phi^H$ where we consider the conjugacy class of $H\leq G$? I tried finding a reference for ...
5
votes
1
answer
742
views
Hypercovers consisting of finite sets
In this paper
on Page 21, the first line of the proof, Peter Scholze seems to claim that any hypercover, consisting of finite sets, splits. I find this hard to believe.
I am not familiar with ...
user473423
46
votes
5
answers
3k
views
‘Naturally occurring’ $K(\pi, n)$ spaces, for $n \geq 2$
[edited!] Given a group $\pi$ and an integer $n>1$, what are examples of Eilenberg–MacLane spaces $K(\pi, n)$ that can be constructed as "known" manifolds? (Or if not a manifold, say some ...
- 2,734
11
votes
1
answer
407
views
Resolutions of unbounded complexes: Condition ($\ast$) in Spaltenstein's paper
In the paper "Resolutions of unbounded complexes" (Compositio Math., vol. 65, no. 2, pp. 121-154) N. Spaltenstein generalizes the 6 functor formalism to unbounded complexes of sheaves over ...
- 23.5k
0
votes
0
answers
220
views
The largest value of $k$ for $\mathbb{Z}^k$ to be embedded in $GL(n,\mathbb{Z})$
This is just a question originated from This conversation (commented by Moishe Kohan). I tried to prove those two assertions but I don't know where to start:
If H is a free abelian subgroup of $SL(n, ...
3
votes
1
answer
160
views
Linking form for homology with general coefficients
For integral homology groups there is the notion of linking form (http://www.map.mpim-bonn.mpg.de/Linking_form)
$$
Tor(H_{l}(X,\mathbb{Z}))\times Tor(H_{n-l-1}(X,\mathbb{Z}))\rightarrow \mathbb{Q}/\...
- 813
8
votes
3
answers
977
views
Examples and properties of spaces with only trivial vector bundles
Let $B$ be a paracompact space with the property that any (topological) vector bundle $E \to B$ is trivial. What are some non-trivial examples of such spaces, and are there any interesting properties ...
- 1,763
2
votes
0
answers
185
views
Properties of universal fibration
I am trying to read the following paper [1] (Becker, James C.; Gottlieb, Daniel Henry
Coverings of fibrations.
Compositio Math.26(1973)) where the authors mentioned that for any fiber $F$,
there ...
- 179
5
votes
1
answer
351
views
"Singular homology = simplicial homology" relative to a fibration
Let $p:E\to B$ be a fibration. Suppose $B$ has a simplicial decomposition. For each $n\in\mathbb{Z}_{\ge0}$, let $C_n$ be the free abelian group generated by the set of pairs $(\sigma,\tau)$ where $\...
- 357
83
votes
7
answers
7k
views
Computational complexity of computing homotopy groups of spheres
At various times I've heard the statement that computing the group structure of $\pi_k S^n$ is algorithmic. But I've never come across a reference claiming this.
Is there a precise algorithm ...
- 44.4k
6
votes
2
answers
438
views
Grouplike and idempotent monoids
Call a monoid group-like if it embeds into its group completion. There exists an obvious tension between group-like and idempotent monoids in that a group cannot contain non-trivial idempotent ...
3
votes
0
answers
429
views
"Maehara-style" proof of Jordan-Schoenflies theorem?
The highest upvoted answer to this old question Nice proof of the Jordan curve theorem? is a proof by Ryuji Maehara. I personally really liked/appreciated that Maehara's proof is
A) a fairly ...
- 833
6
votes
1
answer
478
views
Unbounded acyclic resolutions
Let $\mathscr A$ be a Grothendieck abelian category. Then every object in $\operatorname{Ch}(\mathscr A)$ is quasi-isomorphic to a $K$-injective object [Stacks, Tag 079P]. In particular, for any left ...
- 28.7k
3
votes
1
answer
160
views
Connections on bundle gerbes from cocycle data
I am reading a 2007 article of Bressler et al. on deformation quantization of gerbes. In the article, the authors state that a gerbe on a manifold is defined using certain two-cocycles $c_{ijk}$ ...
- 5,146
8
votes
1
answer
429
views
Gluing of orbifolds
Suppose that $P$ and $Q$ are $n$-dimensional orbifolds, with boundaries. Suppose also that there is an isomorphism $f \colon \partial P \rightarrow \partial Q$ (as orbifolds). Is there a way to glue $...
- 781
3
votes
1
answer
188
views
Removing the symmetry maps from a small category of cubes
Let $[1]^n=\{0<1\}^n$ equipped with the product order. I consider the small category $\widehat{\square}$ of the category of partially ordered sets generated by the coface maps $\delta^\epsilon_i:[1]...
- 3,901
9
votes
1
answer
284
views
Fundamental group of a smoothing of a complex surface
Let $X_0$ be a compact complex algebraic surface with an isolated singularity and let $X_t$ be a smoothing of $X_0$ over the disc. How can we compute the fundamental group of $X_t$ say in terms of the ...
- 3,290
4
votes
2
answers
409
views
Triviality of Steenrod operation on $\Sigma^{2k}\mathbb{CP}^n$
I was going through this paper by Tanaka. I am actually stuck at Lemma 5.2, page 365, given below also
The argument he gives above works, in particular for $\operatorname{Sq}^{2^r-2^j}$ but I am not ...
13
votes
4
answers
2k
views
Fundamental groups of compact Kähler manifolds
This is a sort of a follow-up to this question, and especially to Sean Lawton's answer: The book Fundamental Groups of compact Kähler manifolds (which, in my opinion, is one of the best mathematics ...
- 96.4k
1
vote
0
answers
182
views
Does this sequence stop?
Let $\{ X_i\}$ ($i=1,2,\ldots $) be a family finite CW-complexes such that $X_{i+1}$ is homotopy domintaed by $X_i$, i.e. there exists contionuos maps $g_i:X_i \to X_{i+1}$ and $f_i :X_{i+1} \to X_i$ ...
- 1,182
2
votes
1
answer
174
views
Understanding the picture of monoidal space
Ogus in his slides https://math.berkeley.edu/~ogus/preprints/colloqhandout.pdf presents the following picture of a monoidal space $\operatorname{Spec}(\mathbb{N} \longrightarrow \mathbb{C}[\mathbb{N}])...
- 165
2
votes
1
answer
256
views
Comparing the exit path category and the nerve of a stratified space
Let $P$ be a finite poset, and $X$ be a topological space stratified by $P$, in the sense that $X$ is equipped with a continuous map $X \to P$ in the Alexandroff topology (or equivalently with a ...
- 3,948
2
votes
1
answer
346
views
Calculating degree via homotopy
I'm looking for a reference for the following:
Suppose that $f_1,f_2\colon S^n\rightarrow S^n$ are smooth maps. Let $i\colon S^n\rightarrow \mathbb{R}^{n+1}$ be the inclusion, and suppose that $F\...
- 23
3
votes
0
answers
144
views
Is there a simple explicit expression of the Pontryagin square in terms of the cup product on a spin 4-manifold?
$A$ a finite abelian group, and denote $\Gamma(A)$ its universal quadratic group. The Pontryagin square $\mathfrak{P}\in H^4(B^2A,\Gamma(A))\cong \text{Hom}(\Gamma(A),\Gamma(A))$ is the element ...
- 813
4
votes
0
answers
133
views
Homotopy type / Homology of the free loop space of aspherical manifolds
Let $X$ be a (connected, smooth) closed aspherical manifold. Let $LX:=Map(S^1,X)$ be the free loop space of $X$. Pick $x_0\in X$ and let $\Omega_{x_0}(X)$ be the based loop space of $X$ (based at $x_0$...
- 357
13
votes
3
answers
2k
views
Why do Chern classes and Stiefel-Whitney classes satisfy the "same" Whitney sum formula?
The Whitney sum formula for Stiefel-Whitney classes, $w_n(V \oplus W) = \sum w_i(V) w_{n-i}(W)$, looks a lot like the one for Chern classes $c_n(V \oplus W) = \sum c_i(V)c_{n-i}(W)$. But I don't know ...
- 64k
35
votes
3
answers
1k
views
Second Betti number of lattices in $\mathrm{SL}_3(\mathbf{R})$
We fix $G=\mathrm{SL}_3(\mathbf{R})$.
Let $\Gamma$ be a torsion-free cocompact lattice in $G$. Is $b_2(\Gamma)=0$?
Here the second Betti number $b_2(\Gamma)$ is both the dimension of the ...
- 63.9k
11
votes
1
answer
436
views
A step in Lurie's treatment of $L$-theory
I am looking at Proposition 3 of Lecture 6 from Lurie's course Algebraic L-theory and Surgery (https://www.math.ias.edu/~lurie/287xnotes/Lecture6.pdf). This involves a stable $\infty$-category $\...
- 56.9k
7
votes
1
answer
260
views
Bordism for oriented triangulable manifolds without smooth differentiable structures
We know the bordism group for oriented smooth differentiable structures such as $\Omega_d^{SO}$ that requires the special orthogonal group structure on the tangent bundle $TM$ of manifold $M$.
$$\...
- 10.5k
0
votes
2
answers
545
views
Is $S^1=\mathbb{R}/\mathbb{Z}$ a ring? [closed]
Is there a ring structure on $S^1=\mathbb{R}/\mathbb{Z}$?
- 705
32
votes
3
answers
1k
views
Complex projective manifolds are homeomorphic if homotopy equivalent
If two complex projective manifolds are homotopy equivalent are they homeomorphic?
user164740
13
votes
2
answers
546
views
"Burnside ring" of the natural numbers and algebraic K-theory
The construction of the Burnside ring $A(G)$ of a group $G$ (usually, but not always, finite) is given by taking the Grothendieck group of the commutative semi-ring of isomorphism classes of finite $G$...
- 18.9k