All Questions
9,056 questions
19
votes
1
answer
1k
views
Steenrod squares as power operations vs. as cohomology operations
There are already several excellent questions and answers on MO regarding Steenrod squares, understanding them in various ways and relating them to power operations and I think I get this. Still, I am ...
- 18.4k
3
votes
0
answers
119
views
Signature vs commensurability
If a closed oriented $4n$-dimensional manifold $M$ has an orientation-reversing homeomorphism, then the signature $\sigma(M)$ of the intersection form vanishes. More generally, $\sigma(M) = 0$ if $M$ ...
- 41
3
votes
1
answer
184
views
Freudenthal suspension homomorphism
I asked this question in MathStackExchange a couple of months ago, with little feedback, hence I try here.
The Hopf invariant $h(f)$ of a mapping $f:\mathbb S^{2m-1}\to \mathbb S^m$ is a homotopy ...
- 203
4
votes
2
answers
258
views
Waldhausen S-construction for exact categories
Let $\mathcal{C}$ be an exact category. Then, we can consider $\mathcal{C}$ as a Waldhausen category, where the cofibrations are admissible monomorphisms. By Waldhausen $S$-construction we know that $...
- 189
7
votes
2
answers
323
views
Formula for compositions of Steenrod squares that produce a form in the top degree
On a smooth $d$ dimensional compact connected manifold $M$, for an $\mathbb{Z}_2$-valued $(d-j)$-cocycle $x_{d-j}$ we have the formula ${\text{Sq}}^{j} (x_{d-j}) = u_{j} \cup x_{d-j}$. Here $u_j \in ...
- 10.8k
4
votes
0
answers
177
views
Basis of topology on space of properly embedded smooth manifolds
In A Short Exposition of the Madsen-Weiss Theorem, Hatcher discusses (starting at p.6) a basis for a topology on the space $\mathcal{C}^n$, the space of all smooth oriented $d$-dimensional ...
- 141
2
votes
1
answer
300
views
G-equivariant homotopy between G-spaces
I apologize for asking too many questions in a single post. I am not very conversant with equivariant homotopy theory. While discussing with some faculty I was told that certain fact is true. All ...
- 11.1k
7
votes
2
answers
383
views
Connectivity of fibers under fibration replacement
Assume all the spaces mentioned below are simply connected CW complexes. Let $ f: X \to Y $ be a continuous surjctive map between CW complexes, where $ f $ is not necessarily a fibration. Assume that ...
- 64k
14
votes
1
answer
3k
views
An elementary proof that the degree of a map of spheres determines its homotopy type
I'm helping to teach an undergraduate algebraic topology course (out of Hatcher's textbook). We have recently defined the degree of a map of spheres using homology, and the professor and I thought it ...
- 486
4
votes
1
answer
239
views
True or false? Every left or right cancellative, duo semigroup is cancellative
A semigroup $S$ is duo if $aS = Sa$ for all $a \in S$, where $aS := \{ax: x \in S\}$ and similarly for $Sa$; for instance, every commutative semigroup is duo, and so is every group. On the other hand, ...
- 10.5k
8
votes
1
answer
224
views
Can increasing the winding number of a 2-cell make a CW complex embeddable?
Let $X$ be a 2-dimensional CW complex and let $c\subset X$ be a 2-cell attached along a simple closed loop $\gamma$ in the 1-skeleton $X^{(1)}$.
For a natural number $n\ge 2$ consider the operation of ...
- 976
17
votes
1
answer
414
views
Is $MU/I_\infty$ an $E_\infty$ ring?
Fix a prime $p$, and suppose that $p>2$ for simplicity, although many things should also work for $p=2$. Let $F$ be the usual formal group law defined over $MU_*$, and let $I_\infty$ be the ideal ...
- 52.6k
5
votes
1
answer
378
views
Why is this Brieskorn manifold a rational homology sphere?
In Némethi's book "Normal surface singularities", Example 5.1.17, there is a formula to find the Seifert invariants of a Brieskorn complete intersection $\Sigma(a_1,...,a_n)$. I am ...
- 197
2
votes
1
answer
404
views
Reference request: a cousin to the log semiring
Let $f$ be strictly increasing on $\mathbb{R}$. Then $x \oplus y := f^{-1}(f(x)+f(y))$ gives rise to a strict symmetric monoidal ($\Rightarrow$ commutative monoid) structure on $(\mathbb{R},\ge)$ with ...
CommunityBot
- 1
6
votes
0
answers
128
views
Induced map of degree $k$ self map of a sphere in the higher homotopy groups
Let $f:S^n\rightarrow S^n$ be a degree $k$ map. Then $f$ induces maps $\pi_l(S^n)\rightarrow \pi_l(S^n)$.
I believe that in the stable range ($l\leq 2n-2$) this map is multiplication by $k$. Unstably ...
- 7,583
9
votes
2
answers
1k
views
Number of spin structures
I am probably missing something obvious, but still...
Consider an oriented Riemannian $n$-dimensional vector bundle $\pi: E\rightarrow X$ over compact manifold $X$ with $\omega_2(E)=0$ so it has spin ...
- 95
25
votes
1
answer
582
views
Does every oriented $3$-dimensional submanifold of $\mathbb{R}^6$ bound an oriented $4$-dimensional submanifold?
In my recent research, I encountered the following problem about embeddings. Let $M^3$ be a closed compact oriented smooth $3$-dimensional submanifold of $\mathbb{R}^6$. Does there exist a compact ...
- 161
7
votes
1
answer
297
views
Unoriented cobordism of oriented manifold
We can regard an oriented manifold as an unoriented manifold by forgetting the orientation. This gives a homomorphism from the oriented cobordism group to the unoriented cobordism group. What is the ...
- 1,335
2
votes
0
answers
109
views
Punctured neighbourhood of quotient singularity is not simply connected?
Let $X$ be a variety (irreducible, normal) over a field $k$ which is algebraically closed with characteristic $0$. Suppose that $X$ has only one singular point $p\in X$, so $Y:=X\setminus p$ is smooth....
- 131
1
vote
0
answers
76
views
Pulling back the diagonal class in a Poincaré space
$\DeclareMathOperator{\co}{\operatorname{H}}$Fix a commutative ring $R$. Let $X$ be a connected topological space $X$ which is "$R$-Poincaré of dimension $n$", that is, there exists a (...
- 1,726
4
votes
1
answer
256
views
Third page differential in the Lyndon–Hochschild–Serre Spectral Sequence
I am trying to understand the description of the group cohomology of $Q_8$ from Adem–Milgram’s “Cohomology of finite groups”. The main result is the following:
Theorem 2.9. In the Lyndon–Hochschild–...
- 141
4
votes
1
answer
469
views
How to calculate $\mathrm{TP}(\mathbb{F}_p[t])$?
$\DeclareMathOperator\TP{TP}$I am trying to learn about topological periodic cyclic homology following the notes:
https://www.uni-muenster.de/IVV5WS/WebHop/user/nikolaus/Papers/Lectures.pdf
https://...
- 2,806
16
votes
3
answers
2k
views
When does a CW-complex of dimension 2 embed in $\Bbb R^4$?
Let $X$ be a finite CW-complex of dimension two having just one 0-cell
(+ finitely many 1-cells + finitely many 2-cells).
Is it true that X can be embedded in $\Bbb R^4$?
If true, is it due to ...
- 13.6k
3
votes
1
answer
251
views
About decomposition theorem BBD with respect to some stratification
I want to follow up a question from here (how to deduce version 1.a. from version 1).
I know a version of decomposition theorem BBD:
Version 1. Let $f:X\to Y$ be a (surjective) proper map of complex ...
- 2,806
3
votes
0
answers
181
views
Levelled trees and the homotopy transfer theorem
In section 10.3.12 of Loday-Vallette's book "Algebraic operads", given a $P_\infty$-algebra $(A,d,\alpha)$ the Homotopy Transfer Theorem applied to $H_*(A,d)$ is studied. There, because the ...
- 215
10
votes
2
answers
337
views
Finitely dominated universal spaces for the family of solvable subgroups
$\DeclareMathOperator\PSL{PSL}\DeclareMathOperator\Sz{Sz}$In short, I am interested in the question which finite groups $G$ admit a finitely dominated universal space with respect to the family of ...
- 231
1
vote
0
answers
97
views
Postnikov invariant of crossed square
Is there a reference where Postnikov invariants of the classifying space of a crossed square have been computed ? I am especially interested in the computation of the third Postnikov invariant $B\...
7
votes
1
answer
473
views
Why do homotopy orbits commute with the Eilenberg–Mac Lane spectrum functor?
Let $H: \mathsf{sAb}\to H\mathbb{Z}\text{-}\mathsf{Mod}$ denote the Eilenberg–Mac Lane functor sending a simplicial abelian group $M_\bullet$ to the infinite delooping of its geometric realization, i....
- 56
6
votes
2
answers
523
views
Is the logic of the ($\infty$-?)topos of simplicial sets "contradictory up to homotopy"?
In a sense this is a followup to my earlier question Does the (1-)topos structure on simplicial sets have any homotopy-theoretic significance?.
In the topos of simplicial sets, the subobject ...
- 21.3k
3
votes
0
answers
69
views
How would you call morphisms of varieties that induce isomorphisms on etale cohomology in low degrees?
In our text we have several statements of the following sort: for a certain morphism $f:X\to Y$ of varieties over an (algebraically closed) field of characteristic $p$ and some $c>0$ the ...
- 16.9k
18
votes
7
answers
6k
views
Deeper meanings of barycentric subdivision
I just want to ask if there is any deeper motivation or clear geometric "sense" behind the barycentric subdivision. Some friend asked me about this a few months ago, looking back the section at ...
- 356
6
votes
3
answers
551
views
Conjecture about commutative semigroups
Conjecture: given any commutative semigroup $S$ of order $n \ge 4$, there exist $a, b \in S$ with $a \ne b$, an integer $m \ge \lfloor (n-1)/2 \rfloor$, and two $m$-element subsets $X = \{x_1, \ldots, ...
- 38.6k
8
votes
2
answers
897
views
Can you do geometry with persistent homology?
Setup
In practice, persistent homology of data $X$ is often used to infer the homology of the underlying (Riemannian) manifold $M$ that the data is sampled from.
However most filtrations (Vietoris, ...
- 363
5
votes
1
answer
291
views
Compatibility of natural transformations in a six-functor formalism
Suppose we are given a six-functor formalism and a cartesian diagram
$$\require{AMScd} \begin{CD} X @>\tilde{g}>> Z \\ @V \tilde{f} V V @V Vf V \\ Y @>g>> W\end{CD} \,.$$
There are ...
- 76
7
votes
2
answers
835
views
Holonomy as integration of curvature for principal $G$-bundles?
Holonomy and curvature may seem to be slightly advanced topics in
geometry. However, their origins are easily imaginable. Namely,
picture the surface of earth $S$, and pick an arbitrary
contractible ...
- 5,230
8
votes
1
answer
322
views
Does every cancellative duo semigroup embed into a group?
Prompted by the comments to a recent answer by YCor to a related question (here), I'd like to ask the following:
Q. Does every cancellative duo semigroup embed into a group?
A (multiplicatively ...
- 10.5k
7
votes
2
answers
488
views
Is every cancellative semigroup a subdirect product of subdirectly irreducible cancellative semigroups?
By a classical result of Birkhoff (that is, Theorem 2 in [G. Birkhoff, Subdirect unions in universal algebra, Bull. AMS, 1944]) and the trivial fact that the class of semigroups is closed under the ...
- 10.5k
147
votes
21
answers
23k
views
Are there examples of non-orientable manifolds in nature?
Whilst browsing through Marcel Berger's book "A Panoramic View of Riemannian Geometry" and thinking about the Klein bottle, I came across the sentence:
"The unorientable surfaces are never discussed ...
- 502
2
votes
1
answer
400
views
${\rm SL}_2(\mathbb C)$-equivariant K-theory of $\mathbb C P^1$
Consider the action of ${\rm SL}_2(\mathbb C)$ on $\mathbb C P^1$ induced by the action of 2×2 matrices on 2-vectors. Is it true that $K^{{\rm SL}_2(\mathbb C)}(\mathbb C P^1)$ is $\mathbb C[t,t^{-1}]$...
- 4,111
8
votes
2
answers
596
views
If a semigroup embeds into a group, then is it a subdirect product of groups?
The title has it all:
Q. If a semigroup $S$ embeds into a group, then is $S$ (isomorphic to) a subdirect product of groups?
If yes, then $S$ is a subdirect product of subdirectly irreducible groups,...
- 38.6k
1
vote
1
answer
125
views
Subtlety of identifying $W^{k,p}\bigl([0,1] \bigr)$ and $W^{k,p}(S^1)$ - from ME
I apologize for repeating the same question from ME, but it seems more subtle than I expected.
Let me fix the notations here first:
\begin{equation}
C^\infty_c(0,1):= \{ f : (0,1) \to \mathbb{C} \mid ...
- 3,477
3
votes
0
answers
250
views
Action (of a graded monoid) required
Reference request: Did the construction below appear anywhere before? Any mentions of it or especially any links to something commonly known would be really helpful. I feel that it might be related to ...
- 321
2
votes
0
answers
205
views
What role does homotopy play in Karoubi's K-Theory?
In Karoubi's book K-Theory An Introduction, he defines the groups $K^{p,q}(\mathcal{C})$ for a pseudo-abelian Banach category as equivalence classes of triples $(E,F,\alpha)$, where $E,F \in \mathcal{...
CommunityBot
- 1
2
votes
1
answer
201
views
Identifying $d_1$ in the Atiyah-Hirzebruch-Serre spectral sequence
In A Primer on Spectral Sequences (also later published in More Concise Algebraic Topology), J. Peter May describes the Serre Spectral Sequence for any homology theory. To recap, suppose $p\colon E\...
- 9,263
3
votes
1
answer
327
views
Holomorphic homotopy conjecture
Let $X$ be a smooth projective variety over the complex numbers, and let $\text{Coh}(X)$ be the category of coherent sheaves on $X$. Consider the dg-category $\text{Perf}(X)$ of perfect complexes on $...
- 471
9
votes
0
answers
159
views
Is there a closed aspherical manifold with infinitely many symmetries and without essential immersed tori?
The precise question is the following:
Is there a closed aspherical manifold $M$ of dimension $n\geq 3$ such that Out($\pi_1(M)$) is infinite and $\pi_1(M)$ does not contain $\mathbb Z \times \mathbb ...
- 10.5k
6
votes
0
answers
141
views
Are the $K(n)$-local $E_n$-Adams spectral sequences isomorphic to the Adams-Novikov spectral sequences?
Let $H$ be a closed subgroup of the Morava stabilizer group $\mathbb G_n$. [Devinatz-Hopkins, Prop. 6.7] identifies the $K(n)$-local $E_n$-Adams spectral sequence for $E_n^{hH}$ as the homotopy fixed ...
- 155
4
votes
1
answer
275
views
Why is $bo$ not flat?
Let $bo$ be the connective cover of the real $K$-theory spectrum $KO$. This is a ring spectrum, and so one can look at its Adams spectral sequence. Mahowald does this in "$bo$-resolutions", ...
- 10.8k
2
votes
0
answers
139
views
Is the complement of a square imbedded to a cylinder connected?
Let $C$ be a compact 2-dimensional cylinder $[0,1]\times S^1$. Let $A$, $A'$ be the two connected components of its boundary.
Let $Q$ be a square. Let $a$, $a'$ be a pair of opposite edges of $Q$.
...
- 12.9k
4
votes
1
answer
418
views
Definition of Chow quotient
I am reading M. M. Kapranov's paper "Chow quotients of Grassmannians. I." (English) in Sergej Gelfand (ed.) et al., I. M. Gelfand seminar. Part 2: Papers of the Gelfand seminar in ...
- 4,111