All Questions
9,056 questions
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 ...
- 13.6k
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, ...
- 1,000
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
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
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
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
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
3
votes
1
answer
252
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 ...
- 133
14
votes
1
answer
813
views
What properties of the fundamental group functor are needed to uniquely determine it upto natural isomorphism?
Consider a functor from pointed topological spaces to groups, which
evaluates the same on homotopically equivalent topological spaces and also on homotopic continuous functions.
What additional ...
- 1,525
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 ...
- 913
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
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 $\mathbb C[t,t^{-1}]$...
- 2,964
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", ...
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 ...
- 41
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
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 $...
- 31
4
votes
1
answer
297
views
Fundamental group of the smooth locus of a normal algebraic surface is a quotient of that of a Zariski open subset
Let $X$ be a normal algebraic surface (over $\mathbb{C}$) and $Y$ its smooth locus, i.e., the complement of the singularities of $X$. Suppose $Z\subset Y$ is a Zariski open subset of $X$. Then is it ...
- 335
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
11
votes
2
answers
594
views
In Top, *how* do conjugate homorphisms of groups induce homotopies of classifying maps?
In Top, how do conjugate homorphisms of groups induce homotopies of classifying spaces?
They exist— there's an abstract proof, but how is BG → BH written in terms of the homotopy??
If G and H are only ...
- 301
3
votes
1
answer
351
views
How to define relative orientation in terms of (co)homology?
Let $f\colon X\to Y$ be a smooth surjective map of smooth manifolds of dimension $n$ which are not necessarily orientable. A relative orientation of $X$ over $Y$ consists of an isomorphism $\psi\colon ...
- 3,031
6
votes
1
answer
186
views
Plus construction of the product spaces
I am newly learning plus construction in topology. My question is how to prove the following:
The plus construction of the product of two CW complexes is homotopically equivalent to the product of ...
- 613
5
votes
1
answer
179
views
Euler class in center of mod 2 Morava K-theory?
I consider Morava K-theory at the prime $p=2$ and height $n$. $K(n)^*$ is multiplicative and complex-oriented, but the multiplication is not commutative. Suppose I have a complex bundle $E$ of rank m ...
- 959
6
votes
3
answers
395
views
Decomposable maps of half-smash products
[Cross-posted from MSE]
For a pointed space $X$ and unpointed space $Y$, recall the half-smash product $X\rtimes Y=X\land Y_+=(X\times Y)/(\ast\times Y)$. For unpointed spaces $X,Y$ and a pointed ...
- 508
11
votes
1
answer
690
views
$\zeta(-n)=2^{r_1}\frac{|K_{2n}(O)|}{|K_{2n+1}(O)|} R_K$ and replace $K$-theory with $\mathbb{S}$
There seems to be an agreement among experts that the formulas by Lichtenbaum in the 70's $$\zeta(-n)=2^{r_1}\frac{|K_{2n}(O_K)|}{|K_{2n+1}(O_K)|} R_K$$
follow from the resolution of the Bloch-Kato ...
- 705
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
14
votes
1
answer
573
views
Different proof techniques of the Atiyah-Singer index theorem
I am aware of the usual K-theoretical (cobordism, operator algebras) and heat kernel proofs of the index theorem, as answered in other questions in this site, e.g. here.
However, I recently read this ...
4
votes
1
answer
225
views
Homotopy of Brown-Gitler spectra
Let $A^\vee = \mathbb{F}_2[\bar\xi_1, \bar\xi_2, ...]$ be the mod-2 dual Steenrod algebra. One can define a weight filtration on $A^\vee$ by setting $wt(\bar\xi_i)=2^i$ and $wt(xy)=wt(x)wt(y)$. There ...
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
0
votes
0
answers
61
views
Defining rank of an abelian subgroup using the second centralizer
I recently posted this on MSE, but didn't receive any feedback; so I'm posting it on MO.
I recently came across this article which explored the maximal abelian subgroups of the symmetric group $S_n$. ...
- 195
6
votes
1
answer
313
views
Is Morava K-theory of a classifying space of a compact Lie group a Noetherian ring?
Let $p$ be a prime and $n > 1$ a height. My conventions for Morava K-theory are that $K_p(n)^*(pt)=\mathbb{F}_p[v_n,v_n^{-1}]$, $|v_n|$ (the degree of $v_n$) is $2(p^n-1)$.
Question: If $G$ be a ...
- 3,758
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\...
13
votes
5
answers
2k
views
What are some good examples of spectral sequences which degenerate after the first nontrivial differential?
The difference between algebraic geometry and algebraic topology is that in AG, you usually hope that your spectral sequences degenerate immediately at the $E_2$ page. In AT, you often have to live ...
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\...
- 508
7
votes
1
answer
202
views
Lipschitz bounds and homotopy groups of diffeomorphism groups
Let $M$ denote a closed Riemannian manifold. Let $\mathrm{Diff}_0^L(M)$ denote the supspace of the identity component of the diffeomorphism group $\mathrm{Diff}_0(M)$ of diffeomorphisms with Lipschitz ...
- 1,174
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{...
- 233
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
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
3
votes
1
answer
135
views
Geodesic convexity of Dirichlet Fundamental Domains
My question is motivated by this question, and this answer to it. Below, let's consider the setup in that answer:
Let $M$ be a Riemannian manifold. Let $G\times M\to M$ be a proper action of a ...
- 1,467
4
votes
1
answer
496
views
Homotopy groups of the space of diffeomorphisms
Let $M$ be a smooth, closed, and connected manifold of dimension $n \geq 5$. Let $\operatorname{Diff}(M)$ denote the space of diffeomorphisms of $M$ with the $C^\infty$-topology.
Is there a general ...
- 111
4
votes
1
answer
293
views
Relationship between infinite suspension $\Sigma^{\infty}$ of $E_{\infty}$ grouplike space and its infinite delooping?
For an object $X$ in the infinity category of pointed space $S_{*}$, if it has an $E_{\infty}$ grouplike structure, then it give rises to a unique infinite delooping $BX$, which is a connective ...
- 618
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
7
votes
1
answer
463
views
Direct limits in homotopy category
It is known that the homotopy category $\mathrm{HoTop}$ is not complete nor cocomplete. Moreover, it also fails to have filtered colimits in general. I am wondering if the universal property of the ...
- 173
8
votes
2
answers
723
views
Could there be any homotopy group without "Lebesgue Number Lemma"?
This is about a comment that I have made in my general topology class while I was proving the abovementioned lemma as a consequence of compactness!
As far as I know, essentially, there is only one ...
- 3,173
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 $Q$.
...
- 21.8k
6
votes
1
answer
360
views
On connected sum of compact manifolds along a submanifold
Let $M_1$ and $M_2$ be two compact manifolds of dimension $n\ge 3$. Let us have embeddings $i_1: K \to M_1$ and $i_2: K \to M_2$ for a closed manifold $K$ of dimension at most $n-1$ such that the ...
- 506
13
votes
1
answer
559
views
Intuitive reason for periods of 2 and 8 in Bott periodicity?
Is there a reasonably simple explanation for why Bott periodicity for $U$ and $O$ have periods 2 and 8, respectively? For example, in the $h$-cobordism theorem the requirement that $n \geq 5$ has the ...
- 131
4
votes
1
answer
193
views
Canonical decomposition as wedge sum up to homotopy equivalence
I am curious: is there a canonical way to decompose a finite simplicial complex into a wedge sum up to homotopy equivalence? More formally:
Let $X$ be a finite simplicial complex. Is $X$ homotopy ...
- 8,401
6
votes
0
answers
150
views
Conceptual proof of Jacobi-like identity for Toda brackets
In the paper $p$-primary components of homotopy groups IV, Toda proved an identity for his bracket operation, which can be succinctly written as
$$[[\alpha, \beta, \gamma], \Sigma \delta, \Sigma \...
- 1,263
4
votes
1
answer
184
views
FI-homology of a spectral sequence of rational FI-modules
Let $(E^r_{p,q})$ be a spectral sequence of rational $\mathsf{FI}$-modules. Call $H^{\mathsf{FI}}$ the $\mathsf{FI}$-homology (see here) and $t_k X= \deg H^{\text{FI}}_k X$ the $k$-th generation ...
- 275
9
votes
0
answers
120
views
Reference Request: Moore--Postnikov tower of the rationalization of a fibration
Two spaces $X$ and $Y$ are said to be rationally homotopy equivalent, written $X \sim_{\mathbb{Q}} Y$, if their rationalizations $X_{\mathbb{Q}}$ and $Y_{\mathbb{Q}}$
are homotopy equivalent. Moreover,...
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