All Questions
8,725 questions
16
votes
4
answers
2k
views
Checking if two graphs have the same universal cover
It's possible I just haven't thought hard enough about this, but I've been working at it off and on for a day or two and getting nowhere.
You can define a notion of "covering graph" in graph theory, ...
- 12.6k
7
votes
2
answers
637
views
An algebraic proof of Mumford's smoothness criterion for surfaces?
(Disclaimer: I'm a beginner in this area, so welcome corrections.)
Let $(X,x)$ be a germ of a complex surface (i.e. locally the zero set of some holomorphic functions) and assume that $x$ an isolated ...
- 5,846
18
votes
6
answers
8k
views
PDE on manifolds
I am currently in a PDE course where one of the requirements is to present a paper in PDE. I am wondering if anyone can suggest an early (read foundational, first introductory) paper talking about PDE ...
- 3,726
19
votes
2
answers
1k
views
What manifold has $\mathbb{H}P^{odd}$ as a boundary?
This question is motivated by What manifolds are bounded by RP^odd? (as well as a question a fellow grad student asked me) but I can't seem to generalize any of the provided answers to this setting.
...
- 6,546
6
votes
1
answer
463
views
The (n+1)-st cohomology of K(Z/p,n).
I was looking through my notes for a homotopy theory course and found the following mysterious statement (K is of course the Eilenberg-Maclane space):
$$H^{n+1}(K(\mathbb Z_p,n);\mathbb Z_p) \cong \...
- 3,203
18
votes
12
answers
10k
views
Looking for an introductory textbook on algebraic geometry for an undergraduate lecture course
I am now supposed to organize a tiny lecture course on algebraic geometry for undergraduate students who have an interest in this subject.
I wonder whether there are some basic algebraic geometry ...
25
votes
4
answers
3k
views
A Peculiar Model Structure on Simplicial Sets?
I'm wondering if there is a Quillen model structure on the category of simplicial sets which generalizes the usual model structure, but where every simplicial set is fibrant? I want to use this to do ...
- 27.5k
3
votes
4
answers
658
views
A specific branched cover of S^2 as a subgroup of Pi_1
This is a follow-up question to: Degree 2 branched map from the torus to the sphere
This is a silly computation, but for whatever reason this is taking me much, much longer than it should. So ...
- 1,386
8
votes
5
answers
4k
views
Degree 2 branched map from the torus to the sphere
Algebraic geometry predicts a degree 2 branched cover from an elliptic curve to the projective line. What does this map look like topologically?
- 1,386
36
votes
4
answers
5k
views
Construction of the Stiefel-Whitney and Chern Classes
I've seen two constructions of these characteristic classes. The first comes from Milnor and Stasheff's book and involves the Thom isomorphism and (at least for me) the rather mysterious Steenrod ...
- 3,968
6
votes
4
answers
873
views
Interaction of topology and the Picard group of Algebraic surfaces
It is well known that a smooth cubic surface $X\subset \mathbb{P}^3$ has exactly 27 lines in it. Furthermore, it is easy to check that Picard group $$Pic(X)\cong \mathbb{Z}^7$$ Here the generators are ...
- 2,132
10
votes
1
answer
635
views
Free action of SL_2(F_p) on a sphere
Let $p>2$ be prime. Then for abstract reasons the special linear group $\text{SL}_2({\mathbb F}_p)$ possesses a free action on some sphere (one has to check that any abelian subgroup of $\text{SL}...
- 2,756
3
votes
0
answers
189
views
Which local homeos to numerical space are bijective?
I am reading T. Szamuely's book on Galois groups and fundamental groups.
As preparation to the algebraic case, he recalls the topological case.
So I am wondering if a surjective local homeomorphism $f$...
- 211
22
votes
2
answers
3k
views
Interpretation of elements of H^1 in sheaf cohomology.
Given a variety V and a locally free (coherent) sheaf $\mathcal{F}$ of rank 1 (equivalently a line bundle $L$), I can do a Cech cohomology on it. Then $H^0(V; \mathcal{F})$ are just global sections. ...
- 422
18
votes
1
answer
1k
views
Fundamental groups of the spaces of rational functions
Here is a question which I asked myself (and couldn't answer) while reading "The topology of spaces of rational functions" by G. Segal.
Let $X$ be a smooth complete complex curve (=a compact Riemann ...
- 23.5k
11
votes
1
answer
671
views
Is it always possible to compute the Betti numbers of a nice space with a well-chosen Lefschetz zeta function?
Let $X$ be a smooth projective variety. If I've understood correctly, the Weil conjectures imply that it is possible to compute the Betti numbers of $X(\mathbb{C})$ by computing the local zeta ...
- 118k
6
votes
2
answers
2k
views
Understanding the product in topological K-theory
I apologize that this is perhaps not adequate for mathoverflow but I have struggled with this for days now and become desperate...
The reduced K-group $\tilde{K}(S^0)$ of the zero sphere is the ring $...
- 2,782
11
votes
1
answer
604
views
Do h-coequalizers and coproducts give all h-colimits?
It is well known that if a category has all coequalizers and all (small) coproducts then in fact it has all (small) colimits. More important is the proof which shows that every colimit can be built by ...
- 27.5k
31
votes
4
answers
8k
views
What is 'formal' ?
The key step in Kontsevich's proof of deformation quantization of Poisson manifolds is the so-called formality theorem where 'a formal complex' means that it admits a certain condition. I wonder why ...
- 571
12
votes
1
answer
651
views
Does a triangulation without fixed simplex property always exist?
Suppose we are given a triangulable topological space $X$. If $X$ has the fixed point property (FPP), then obviously for every triangulation $K$ of $X$ and every simplicial map $f:K\to K$ a simplex $\...
- 2,104
6
votes
2
answers
985
views
Lifting a homeomorphism, always possible?
Let $h:M\to M$ be a homeomorphism of a compact manifold. Let $p:\tilde M\to M$ be a covering. 1) Is it always possible to lift $h$ to $H:\tilde M\to \tilde M$ so that everything fits into the ...
- 1,414
26
votes
2
answers
2k
views
Loop Spaces as Generalized Smooth spaces or as Infinite dimensional Manifolds?
There are two ways to define smooth mapping spaces and I want to know how they compare.
Let's take the concrete special case of free loops spaces. I think this is the most studied example so will ...
- 27.5k
4
votes
3
answers
442
views
Are there universe-indexed spectra over simplicial sets?
In "Rings, Modules, and Algebras in Stable Homotopy Theory" Elmendorf, Kriz, Mandell and May introduce a notation of spectra indexed over an universe $\mathcal{U}$ as a collection of pointed ...
- 1,273
20
votes
1
answer
8k
views
Universal Covering Space of Wedge Products
Today I was studying for a qualifying exam, and I came up with the following question;
Is there a simple description in terms of the subspaces universal covers for the universal cover of a wedge ...
- 4,842
3
votes
1
answer
978
views
cell complexes and higher graph theory
Suppose that, on an intuitive basis, one defines a "2-graph" $(V,E,F,\partial)$ as a collection of vertices, oriented edges and oriented faces, all of which should be considered as abstract objects ...
- 503
22
votes
7
answers
3k
views
Essential theorems in group (co)homology
I'm trying to fill in the gaps in my understanding of group (co)homology and I'm wondering what are considered the "must know" theorems and concepts. I'm thinking of things along the lines of
Hopf's ...
3
votes
1
answer
225
views
Explicit classifying spaces for crossed complexes
I'm trying to understand the topology behind a certain group which fits into a truncated crossed complex, so I've been trying to understand Brown's construction of the classifying space of a crossed ...
- 1,422
17
votes
4
answers
3k
views
Group Structure on CP^infinty
I was inspired by the following algebraic topology orals question:
"Is $S^1$ the loop space of another space?"
This is easy to see if you recognize that $S^1$ is a $K(\mathbb{Z},1)$, and the loop ...
- 2,684
7
votes
2
answers
1k
views
Maximal Ellipsoid
John's Theorem can be stated as "To every compact, convex body, there is a unique inscribed ellipsoid, whose volume is maximal among all inscribed ellipsoids." It goes on to classify this maximal ...
- 1,588
4
votes
1
answer
1k
views
references for models of stable infinity categories
There's a fair amount of literature comparing different models for the homotopy theory of homotopy theories, or the homotopy theory of $(\infty,1)$-categories. Julie Bergner has a survey of this ...
- 293
11
votes
2
answers
2k
views
Meaning of orientation/orientability over rings other than the integers
This was asked as part of an earlier question. But since this part did not attract many answers, I am asking it separately.
We consider the homology definition of an orientation for a manifold, as ...
- 7,442
4
votes
2
answers
592
views
Five lemma in HoTop* and arbitrary pointed model categories
Let $\textbf{HoTop}^*$ be the homotopy category of pointed topological spaces. In the following, the word "isomorphism" shall always mean isomorphism in $\textbf{HoTop}^*$, i.e. pointed homotopy ...
- 2,756
28
votes
4
answers
4k
views
Classifying Space of a Group Extension
Consider a short exact sequence of Abelian groups -- I'm happy to assume they're finite as a toy example:
$$
0 \to H \to G \to G/H \to 0\ .
$$
I want to understand the classifying space of $G$. Since ...
- 4,217
26
votes
3
answers
2k
views
Reverse mathematics of (co)homology?
Background
Exercise 2.1.16b in Hartshorne (homework!) asks you to prove that if $0 \rightarrow F \rightarrow G \rightarrow H \rightarrow 0$ is an exact sequence of sheaves, and F is flasque, then $0 \...
- 12.1k
9
votes
2
answers
744
views
Cobordisms of bundles?
Is there a notion of a cobordism which is compatible with bundle structure?
That is, if I have bundles $E$ and $F$, is it the case that the manifold $W$ with $E$ and $F$ as boundary components, can ...
- 2,179
3
votes
1
answer
166
views
Upper bound on the genus of a k-page graph
Is there an upper bound on the genus of a graph that has a book embedding on say k pages, or can the genus be arbitrarily large? If not a general bound is known, what happens for k=3?
- 75
17
votes
4
answers
2k
views
What is the Euler characteristic of a Hilbert scheme of points of a singular algebraic curve?
Let $X$ be a smooth surface of genus $g$ and $S^nX$ its n-symmetrical product (that is, the quotient of $X \times ... \times X$ by the symmetric group $S_n$). There is a well known, cool formula ...
- 445
5
votes
2
answers
454
views
Burnside ring and zeroth G-equivariant stem for finite G
Let $G$ be a finite group. The theorem that the Burnside ring $A(G)$ is isomorphic to the zeroth stable stem $\pi^{G}_0(S)$ is usually said to originate from Segal. I search for a reference of a proof ...
- 1,273
16
votes
7
answers
2k
views
two conjugate subgroups and one is a proper subset of the other? plus, a covering space interpretation.
Recently I've been reading J.P. May's A Concise Course in Algebraic Topology. In the section on the classification of covering groupoids, he mentions that sometimes a group G may have two conjugate ...
- 6,069
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 ...
7
votes
1
answer
458
views
Reference for equivalent definitions of the genus
Let $X$ be a (edit: nonsingular) projective complex algebraic curve. The genus of $X$ can be defined as the dimension of the space of holomorphic $1$-forms on $X$, which in turn can be defined either ...
26
votes
2
answers
5k
views
Cohomology of Lie groups and Lie algebras
The length of this question has got a little bit out of hand. I apologize.
Basically, this is a question about the relationship between the cohomology of Lie groups and Lie algebras, and maybe ...
- 23.5k
14
votes
1
answer
4k
views
complex structure on S^n
Using the chern character, it can be shown that there is no complex structure on $S^n$ if $n > 6$: See May's book: if $S^{2n}$ has a complex structure, let $\tau$ be the tangent bundle. $c_n(\tau) =...
user19475
9
votes
3
answers
4k
views
Grassmannian bundle theorem
Let's consider a vector bundle $E$ of rank $n$ over a compact manifold $X$. Consider the associated Grassmannian bundle $G$ for some $k < n$, obtained by replacing each fiber $E_x$ by $Gr(k,E_x)$.
...
- 1,783
6
votes
2
answers
657
views
Properties of the class of topological spaces possessing a CW-structure
Let ${\mathcal C}$ be the class of topological spaces which carry a CW-structure (note that I do not want to fix some particular CW-structure).
Is it true that for a covering map $E\stackrel{f}{\to} ...
- 2,756
16
votes
12
answers
10k
views
How seriously should a graduate student take teaching evaluations? [closed]
Pretty much the question in the title. If a grad student gets bad reviews as a TA, how much does that hurt them later? How much do good reviews help? What if the situation is more complex? (For ...
17
votes
4
answers
2k
views
Applications of the Brown Representability Theorem
Probably you can "google" this question, but I can't find anything relevant. The classical Brown Representability Theorem states: Denote $hCW_*$ the homotopy category of pointed CW-complexes. Let $F : ...
- 63.1k
14
votes
2
answers
1k
views
Unpointed Brown representability theorem
The classical Brown Representability Theorem states: Denote $hCW_*$ the homotopy category of pointed CW-complexes. Let $F : hCW_* \to Set_*$ be a contravariant functor. Then $F$ is representable if ...
- 63.1k
21
votes
3
answers
2k
views
Cohomology of fibrations over the circle: how to compute the ring structure?
This question is inspired by Cohomology of fibrations over the circle Moreover, it can be considered a subquestion of the above, but somehow it seems to me that some of the more interesting points ...
- 23.5k
30
votes
3
answers
3k
views
Mumford conjecture: Heuristic reasons? Generalizations? ... Algebraic geometry approaches?
The Mumford conjecture states that for each integer $n$, we have: the map $\mathbb{Q}[x_1,x_2,\dots] \to H^\ast(M_g ; \mathbb{Q})$ sending $x_i$ to the kappa class $\kappa_i$, is an isomorphism in ...
- 21k