All Questions
1,240 questions
6
votes
1
answer
954
views
What is a higher derived constructible sheaf
Suppose $X$ is a topological space and $k$ some discrete coefficient field. Let's define the category of "$\infty$-local systems on $X$" to be DG representations of the ring $C_*(\Omega X,k)$ of ...
- 7,962
6
votes
0
answers
209
views
If $X_d$ is a non-triangulable manifold, can $X_d \times T^k$, $X_d \times I^k$, or $X_d \times \mathbb{R}^k$ be a triangulable manifold?
If $X_d$ is a non-triangulable manifold, can $X_d \times T^k$, $X_d \times I^k$, or $X_d \times \mathbb{R}^k$ always be a triangulable manifold?
Let $X_d$ be a $d$-manifold which is NOT a ...
- 10.5k
6
votes
2
answers
2k
views
Reference for a fact (?) on homeomorphic knot complements
Does somebody have a reference (or an argument why it should be true) for the following statement?
“Let $K$ and $K'$ be knots in $S^3$. If there is an orientation-preserving homeomorphism $h : S^3 \...
- 63
6
votes
1
answer
1k
views
Homotopy pullbacks of simplicial spaces, and Bousfield-Friedlander
Let $X_\bullet \longrightarrow Y_\bullet \longleftarrow Z_\bullet$ be a diagram of simplicial spaces (=bisimplicial sets, if you like).
On p. 14-9 of these notes there is an example which shows that ...
- 18.2k
6
votes
3
answers
2k
views
Mayer-Vietoris homotopy groups sequence of a pull-back of a fibration
This must be an elementary question: could somebody tell me a reference for the Mayer-Vietoris homotopy groups sequence of a pull-back of a fibration?
I'm working in the category of pointed ...
- 1,975
5
votes
1
answer
843
views
When is homotopy orbit space weakly equivalent to orbit space, other than situation of free action?
Let $M$ be a closed symmetric monoidal model category. Let $X$ be a cofibrant object (it can also be fibrant if you like) and let $\Sigma_n$ act on $X^{\otimes n}$ by permuting the factors (note that ...
- 30.3k
5
votes
2
answers
2k
views
Canonical reference for Chern characteristic classes
I'm a little uncertain about the definitions for
Chern roots
Chern classes
Chern characters
From perusing several discussions, I gather that if one correlates the nomenclature with that of ...
- 10.5k
5
votes
1
answer
602
views
Are there any fake 3-tori?
Hsiang, W.-c.; Shaneson, J. L. Fake tori, the annulus conjecture, and the conjectures of Kirby. Proc. Nat. Acad. Sci. U.S.A. 62 1969 687–691.
The paper above classified all fake tori for dimension $\...
- 2,623
5
votes
3
answers
2k
views
Continuous change of basis (and on the definition of determinant) [closed]
Let $(u_1, \ldots, u_n)$ and $(v_1, \ldots, v_n)$ be two ordered bases of $\mathbb R^n$. The orientation of the first basis is defined as the sign of the determinant of $[u_1 \cdots u_n]$, and ...
- 1,174
5
votes
1
answer
1k
views
Bott's Formula for Grassmannians
Bott's Formula gives the dimension of the cohomology $H^{q}(\mathbb{P}^{n}, \Omega_{\mathbb{P}^{n}}^{p}(k))$ of the $k$-twisted sheaf of $p$-differential forms on the projective space $\mathbb{P}_{\...
- 283
5
votes
0
answers
248
views
Homotopy theory of cospaces (or $\infty$-cogroupoids)
Is there a good homotopy theory for cospaces, where a cospace (or $\infty$-cogroupoid) would be a cosimplicial set satisfying some appropriate dual version of the Kan condition?
One point I'm curious ...
- 11.8k
5
votes
1
answer
503
views
Every unorientable 4-manifold has a $Pin^c$, $Pin^{\tilde c+}$ or $Pin^{\tilde c-}$ Structure
The precise statement on J. W. Morgan's "The Seiberg-Witten Equations and Applications to the Topology of Smooth Four-Manifolds (MN-44)" that 4-manifold $X$ admits a Spinc structure (Lemma 3.1.2) ...
- 10.5k
5
votes
1
answer
541
views
Complexes of sheaves with locally constant cohomology versus $C_{*}(\Omega M)$-modules
Let $M$ be a nice, connected topological space. Assume it is a manifold, if you like.
There are two rather similar looking differential-graded (dg) categories that one can associate to $M$ that ...
- 51
5
votes
2
answers
2k
views
Homotopy limit-colimit diagrams in stable model categories
It is shown in Remark 7.1.12 of (a newer version of) Mark Hovey's book Model Categories that, in a stable model category, homotopy pullback squares coincide with homotopy pushout squares. The argument ...
- 3,184
4
votes
1
answer
329
views
Bredon cohomology of $\mathbb{S}^\sigma$
I tried to compute Bredon cohomology of $\mathbb{S}^\sigma$, where $\sigma$ is a sign representation of $\mathbb{Z}/2$, following first chapter and first construction of cohomology from Bredon's "...
- 1,759
4
votes
1
answer
361
views
Model structure on category of endofunctors
Let $\mathcal C$ be model category, perhaps even cofibrantly generated. I don't assume that $\mathcal C$ is small. Recall that $End(\mathcal C)$ is the category of endofunctors on $\mathcal C$, with ...
- 583
4
votes
1
answer
188
views
Is there an elementary subexponential upper bound on the size of the stable stems?
This is a question in stable homotopy theory which I will boil down to a pure combinatorics question. If you're not interested in the homotopy theory, feel free to skip to the end for the ...
- 64k
3
votes
1
answer
441
views
In which topological spaces does the existence of a loop not contractible to a point imply there is a non-contractible simple loop also?
In another MathOverflow post I asked: In a topological space if there exists a loop that cannot be contracted to a point does there exist a simple loop that cannot be contracted also?
Note that ...
- 4,862
3
votes
1
answer
241
views
Restricting spectra to finite $n$-truncated/$n$-connected pointed spaces
$\renewcommand{\S}{\mathcal{S}}\newcommand{\l}{\langle}\newcommand{\r}{\rangle}\newcommand{\op}{\mathsf{op}}\newcommand{\fin}{\mathrm{fin}}$Recently I've noticed that the definitions of special $\...
- 11.8k
3
votes
3
answers
1k
views
Classifying spaces, Brown representability, and homotopy equivalences
Let $G_1$ and $G_2$ be topological groups. Assume that there exists a continuous homomorphism $f : G_1 \rightarrow G_2$ which (ignoring the group structure) is a homotopy equivalence. If $BG_i$ is a ...
- 33
2
votes
0
answers
429
views
A Generalized De Rham cohomology
Edit According to the comment of Alex Degtyarev, I deleted the last part of the previous version.
Let $E$ be a real vector space. The complex valued $k$- tensors on $E$ is denoted by $L_{\mathbb{C}}^{...
- 366
2
votes
0
answers
127
views
If $\Omega f$ is an $E_\ast$-equivalence, then is $\Omega^2 f$ an $E_\ast$-equivalence?
Let $E_\ast$ be a homology theory and $f: X \to Y$ a map of spaces. Suppose that $\Omega f: \Omega X \to \Omega Y$ is an $E_\ast$-equivalence. Then is $\Omega^2 f: \Omega^2 X \to \Omega^2 Y$ an $E_\...
- 64k
2
votes
0
answers
165
views
Automorphism group of a Lie group $G$ vs that of a covering group $\tilde G$: same or not?
Is it true or false that the Inner (Inn), Outer (Out) and Total (Aut) Automorphism of a Lie group $G$ is the same as the covering group of the Lie group, say $\tilde G$ (regardless of how many types ...
- 10.5k
2
votes
1
answer
359
views
Induced Map on Sp(2g,Z) is surjective
Let Mg be the Mapping Class Group for $S_g$, the genus-g orientable surface, and consider the action of Mg on $H_1(S_g,\mathbb Z)$ sending f in Mg to m in $Sp^2(2g,\mathbb Z)$ through the induced map ...
- 105
2
votes
0
answers
82
views
Enveloping a Jordan curve with a trace of another one
This question is inspired by this one, or rather the way I understood it.
Let $\gamma$ and $\delta$ be parametrised Jordan curves on the plane (i.e. homeomorphisms from $S^1$ onto its image in $\...
- 5,529
1
vote
1
answer
389
views
About isotopy of simple close curve
In the Primer mapping class group by farb Margalit. We have :
Proposition 1.10 Let $\alpha$ and $\beta$ be two essential simple closed curves in a surface $S$. Then $\alpha$ is isotopic to $\beta$ if ...
- 119
1
vote
0
answers
131
views
Can we construct a general counterexample to support the weak whitney embedding theorm?
The weak Whitney embedding theorem states that any continuous function from an $n$-dimensional manifold to an $m$-dimensional manifold may be approximated by a smooth embedding provided $m > 2n$.
...
- 109
-1
votes
1
answer
209
views
A closed manifold with a subset with the same ring cohomology
Is there an example of a closed manifold $M$ with a proper subset $A\subset M$ such the inclusion $i:A \to M$ gives a ring isomorphism $i^{*}$ between $\mathbb{Z}$-cohomologies?
In this question $...
- 366
158
votes
8
answers
7k
views
Resources for mathematics advising.
This question is possibly ill-advised. (If it is not right for this site I will delete it.)
I, suddenly, have students.
It is very clear to me that there is nothing in my education that has ...
152
votes
13
answers
22k
views
Why is the fundamental group of a compact Riemann surface not free ?
Consider a compact Riemann surface $X$ of genus $g$.
It is well-known that its fundamental group $\pi_1(X)$ is the free group on the generators $a_1,b_1,...,a_g,b_g$ divided out by the normal ...
- 47.5k
141
votes
0
answers
13k
views
Grothendieck-Teichmüller conjecture
(1) In "Esquisse d'un programme", Grothendieck conjectures
Grothendieck-Teichmüller conjecture: the morphism
$$
G_{\mathbb{Q}} \longrightarrow Aut(\widehat{T})
$$
is an isomorphism.
Here $...
- 7,527
114
votes
34
answers
86k
views
Why do we teach calculus students the derivative as a limit?
I'm not teaching calculus right now, but I talk to someone who does, and the question that came up is why emphasize the $h \to 0$ definition of a derivative to calculus students?
Something a teacher ...
97
votes
19
answers
38k
views
Collecting proofs that finite multiplicative subgroups of fields are cyclic
I teach elementary number theory and discrete mathematics to students who come with no abstract algebra. I have found proving the key theorem that finite multiplicative subgroups of fields are cyclic ...
93
votes
3
answers
11k
views
What is homology anyway?
Disclaimer: I don't feel qualified to ask this question and yet it's been troubling me for some time now and I lost my patience and decided to ask to get some kind of answer. If there are any stupid ...
- 7,799
87
votes
11
answers
14k
views
What is Quantization ?
I would like to know what quantization is, I mean I would like to have some elementary examples, some soft nontechnical definition, some explanation about what do mathematicians quantize?, can we ...
83
votes
0
answers
3k
views
Which finite abelian groups aren't homotopy groups of spheres?
Someone asked me if all finite abelian groups arise as homotopy groups of spheres. I strongly doubted it, and I bet ten bucks that $\mathbb{Z}_5$ is not $\pi_k(S^n)$ for any $n,k$. But I don't know ...
- 22.3k
72
votes
3
answers
8k
views
Where do all these projection formulas come from?
I have been intrigued for a long time by the formal similarity of results from different areas of mathematics. Here are some examples.
Set theory Given a map $f:X\to Y$ and subsets $X' \subset X, Y'\...
- 47.5k
69
votes
20
answers
19k
views
Fun applications of representations of finite groups
Are there some fun applications of the theory of representations of finite groups? I would like to have some examples that could be explained to a student who knows what is a finite group but does not ...
66
votes
5
answers
8k
views
Does homology have a coproduct?
Standard algebraic topology defines the cup product which defines a ring structure on the cohomology of a topological space. This ring structure arises because cohomology is a contravariant functor ...
- 661
66
votes
1
answer
2k
views
Is there an octonionic analog of the K3 surface, with implications for stable homotopy groups of spheres?
The infamous K3 surface has many constructions in many fields ranging from algebraic geometry to algebraic topology. Its many properties are well known. For this question I am really interested in the ...
- 27.5k
63
votes
0
answers
2k
views
Are there periodicity phenomena in manifold topology with odd period?
The study of $n$-manifolds has some well-known periodicities in $n$ with period a power of $2$:
$n \bmod 2$ is important. Poincaré duality implies that odd-dimensional compact oriented manifolds ...
- 118k
61
votes
2
answers
3k
views
The topological analog of flatness?
Recall that a map $f:X\to Y$ of schemes is called flat iff for any $x\in X$ the ring $O_{X,x}$ is a flat $O_{Y,f(x)}$-module.
Briefly the question is: what is the topological analog of this?
Many ...
- 23.5k
58
votes
4
answers
5k
views
Advice for PhD Supervisors
My first PhD student is having his viva tomorrow. Hence, I began contemplating a bit about the whole process of supervising. One thing I realized is that while there seems to be plenty of advice for ...
56
votes
2
answers
3k
views
How to add essentially new knots to the universe?
A knot is an embedding of a circle $S^{1}$ in $3$-dimensional Euclidean space, $\mathbb{R}^3$. Knots are considered equivalent under ambient isotopy. There are two different types of knots, tame and ...
54
votes
6
answers
8k
views
"Why the heck are the homotopy groups of the sphere so damn complicated?"
This is a quote from a dear friend asking the rest of us on Facebook. I gave him some half-baked response, but the truth is I don't really know enough about this to give him a good response.
So why ...
54
votes
4
answers
9k
views
Why is Quantum Field Theory so topological?
I understand that my question suffers from my lack of knowledge about the field, but as a mathematician without much knowledge of physics I have been wondering much about the following and I always ...
53
votes
7
answers
8k
views
Zorn's lemma: old friend or historical relic?
It is often said that instead of proving a great theorem a mathematician's fondest dream is to prove a great lemma. Something like Kőnig's tree lemma, or Yoneda's lemma, or really anything from this ...
- 18.7k
53
votes
4
answers
14k
views
Explanation for the Chern character
The Chern character is often seen as just being a convenient way to get a ring homomorphism from K-theory to (ordinary) cohomology.
The most usual definition in that case seems to just be to define ...
- 5,530
53
votes
6
answers
8k
views
Why is the standard definition of cocycle the one that _always_ comes up??
This question might not have a good answer. It was something that occurred to me yesterday when I found myself in a pub, needing to do an explicit calculation with 2-cocycles but with no references ...
- 41.4k
52
votes
22
answers
19k
views
Interesting Calculus Questions/Exercises
I am in the process of redesigning the calculus course that I have taught five or six times. What I would like to know is if anyone has some really good examples or exercises that I could either do ...