All Questions
8,725 questions
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
14
votes
1
answer
816
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 ...
- 1,545
3
votes
1
answer
253
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
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
5
votes
1
answer
294
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
401
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
276
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
328
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 $...
- 41
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
11
votes
6
answers
2k
views
Hard problems with an easy-to-understand answer
I am very interested by problem in mathematics which are difficult (go at least 10 years without a resolution, say) but which have a solution that is short and elementary.
In this video Launay gave an ...
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
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
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
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
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
188
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
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
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
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
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
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 ...
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\...
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
160
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
294
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
1
vote
0
answers
106
views
The proposition associated with a set
Given a set $U$ and a set $A \subseteq U$, is there an accepted symbol for the proposition $p$ over the universe $U$ such that for each $x \in U$, $p(x)$ is the assertion that $x \in A$? (The symbol $...
- 19.7k
7
votes
1
answer
464
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
730
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
560
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
$$[[\alpha, \beta, \gamma], \Sigma \delta, \Sigma \...
- 1,262
4
votes
1
answer
184
views
FI-homology of a spectral sequence of rational FI-modules
Let $(E^r_{p,q})$ be a spectral sequence of rational $\mathsf{FI}$-modules. Call $H^{\mathsf{FI}}$ the $\mathsf{FI}$-homology (see here) and $t_k X= \deg H^{\text{FI}}_k X$ the $k$-th generation ...
- 275
34
votes
6
answers
4k
views
Why study finite topological spaces?
In rereading Thurston's essay On Proof and Progress in Mathematics I ran across this passage:
… this means that some concepts that I use freely and naturally in
my personal thinking are foreign to ...
- 737
5
votes
1
answer
222
views
Serre spectral sequence of Borel construction
Let $G$ be finite $p$-group, $X$ be path-connected $G$-space, $E=EG\times_{G}X$ be the Borel construction and $BG$ be the classifying space of $G$. Consider Serre spectral sequence of the fibration
$$...
- 51
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,...
- 371
7
votes
1
answer
435
views
What are the covering spaces of $\mathbb{Q}$?
Let $X = \mathbb{Q}$, topologized as a subset of the real line. Is there a reasonable description of the covering spaces of $X$?
Here is something more precise. One way of constructing covers $p: \...
- 395
28
votes
3
answers
1k
views
Proofs of Poincaré duality
I know several proofs of Poincaré duality:
The original proof using dual cell complexes. Probably the nicest version of this uses a handle decomposition.
The argument (in Hatcher and many other ...
- 281
0
votes
0
answers
92
views
About filtration of the Leray-Serre spectral sequence
In the following proof, it is used the spectral sequence of the Borel
fibration $X\longrightarrow X_{T}\longrightarrow B_{T}$.
I don't understand how the map $\psi $ is obtained, and how is it an $R$-...
- 1,367