All Questions
1,240 questions
25
votes
3
answers
3k
views
Non trivial vector bundle over non-paracompact contractible space
The proof that the set of classes of vector bundles is homotopy invariant relies on the paracompactness and the Hausdorff property of the base space. Are there any known examples of:
Non trivial ...
- 281
25
votes
19
answers
20k
views
Math books for advanced high school students
I'm working in a program for teaching a group of students selected in a Olympiad competition. The program is aimed to acquaint the students with the diverse aspects of higher mathematics in a way ...
24
votes
1
answer
1k
views
Combinatorial spin structures
I would like to know how to define spin structures combinatorially, for an oriented smooth manifold equipped with a triangulation. In the case of a 2d manifold, spin structures correspond to ...
- 1,579
24
votes
1
answer
467
views
To what extent can we characterise the image of the topological Chern character?
For a finite CW complex $X$, the Chern character gives an isomorphism
of finite-dimensional vector spaces:
$$
ch : K^*(X)\otimes \mathbb{Q} \to H^*(X, \mathbb{Q}).
$$
The vector space $V = H^*(X, \...
- 1,444
24
votes
2
answers
2k
views
Direct proof that the centralizer of $GL(V)$ acting on $V^{\otimes n}$ is spanned by $S_n$
Let $V$ be a finite dimensional vector space over a field of characteristic zero. Let $A$ be the space of maps in $\mathrm{End}(V^{\otimes n})$ which commute with the natural $GL(V)$ action. Clearly, ...
- 156k
24
votes
2
answers
2k
views
Good functorial model for BG
There are several functorial constructions of the space BG for a topological group (meaning BG plus the universal G-bundle). First, there is the Milnor construction, treated in several textbooks. The ...
- 20.9k
24
votes
4
answers
3k
views
What do cohomology operations have to do with the non-existence of commutative cochains over $\mathbb{Z}$?
Let $X$ be a topological space. In elementary algebraic topology, the cup product $\phi \cup \psi$ of cochains $\phi \in H^p(X), \psi \in H^q(X)$ is defined on a chain $\sigma \in C_{p+q}(X)$ by $(\...
- 25.6k
24
votes
10
answers
4k
views
Why localize spaces with respect to homology?
A basic construction in algebraic topology is the localization of spaces or spectra with respect to a homology theory: one formally inverts the $E$-homology isomorphisms, reflecting each space into ...
- 66.8k
24
votes
3
answers
2k
views
Are there topological obstructions to the existence of almost quaternionic structures on compact manifolds?
$\DeclareMathOperator\End{End}\newcommand\Id{\mathrm{Id}}\DeclareMathOperator\Sp{Sp}\DeclareMathOperator\SO{SO}$I start with some background, but people familiar with the subject may jump directly to ...
- 2,140
24
votes
2
answers
3k
views
Are homeomorphic open subsets of $\mathbb{R}^n$ also diffeomorphic?
Let $U_1, U_2$ be open subsets of $\mathbb{R}^n$. Both are naturally differentiable submanifold, getting the differentiable structure from $\mathbb{R}^n$. Further, both are natural topological ...
- 243
24
votes
2
answers
3k
views
Roadmap to Hill-Hopkins-Ravenel
How does one go from an understanding of basic algebraic topology (on the level of Allen Hatcher's Algebraic Topology and J.P. May's A Concise Course in Algebraic Topology) to understanding the paper ...
- 3,309
24
votes
2
answers
4k
views
Isotopy extension theorems
I'm looking for the origins of the isotopy extension theorem in categories other than the smooth category.
Precisely, in the smooth category, the isotopy extension theorem says that if $f : [0,1] \...
- 44.4k
24
votes
1
answer
5k
views
Do "surjective" degree zero maps exist?
Is there a map $f\colon X \to Y$ of closed, connected, smooth and orientable $n$-dimensional manifolds such that the degree of $f$ is 0 but $f$ is not homotopic to a non-surjective map?
Added: The ...
- 2,590
24
votes
3
answers
2k
views
Generalization of Borsuk-Ulam
Let $n$ be a positive intger. Is the following true? For continuous maps $f: \mathbb S^n \rightarrow \mathbb S^n$ and $g: \mathbb S^n \rightarrow \mathbb R^n$, there exists a point $x \in \mathbb S^n$ ...
- 11.9k
24
votes
9
answers
9k
views
How to motivate and present epsilon-delta proofs to undergraduates?
This would seem to be a common question, but I am surprised not to see it already asked and answered on MO!
I am teaching an undergraduate course, and I want to teach them to construct basic epsilon-...
24
votes
4
answers
2k
views
How many simplicial complexes on n vertices up to homotopy equivalence?
Fix a number $n$, and define $\gamma(n)$ to be the number of simplicial complexes on $n$ unlabeled vertices up to homotopy equivalence. It is unlikely that an explicit formula exists, but what is ...
- 15.5k
24
votes
0
answers
1k
views
p-Adic String Theory and the String-orientation of Topological Modular Forms (tmf)
I am going to ask a question, at the end below, on whether anyone has tried to make more explicit what should be, it seems to me, a close relation between p-adic string theory and the refinement of ...
- 19.9k
24
votes
1
answer
968
views
Groups whose finite index subgroups of fixed index are isomorphic
I am interested in finitely generated groups $G$ that are residually finite and have the following property: For each $d \geq 1$, $G$ has subgroups of finite index $d$, and all such subgroups are ...
- 11.9k
24
votes
3
answers
4k
views
Is there a map of spectra implementing the Thom isomorphism?
A well known theorem in algebraic topology relates the (co)homology of the Thom space $X^\mu$ of a orientable vector bundle $\mu$ of dimension $n$ over a space $X$ to the (co)homology of $X$ itself: $...
- 8,167
23
votes
0
answers
590
views
What is the symmetric monoidal functor from Clifford algebras to invertible K-module spectra?
There ought to be a symmetric monoidal functor from the symmetric monoidal $2$-groupoid whose objects are Morita-invertible real superalgebras (precisely the Clifford algebras), morphisms are ...
- 118k
23
votes
3
answers
2k
views
A homology theory which satisfies Milnor's additivity axiom but not the direct limit axiom?
Let us agree on the following: a "homology theory" means a functor $h_*$ from the category of pointed CW complexes to the category of graded abelian groups, together with natural isomorphisms $h_{*+1}(...
- 3,004
23
votes
3
answers
3k
views
Brauer Groups and K-Theory
Is there some a priori reason why we should expect the Brauer group of real [complex] super vector spaces to be closely related to periodicity in real [complex] K-theory? By "a priori" I mean a proof ...
- 12.8k
23
votes
5
answers
5k
views
Stiefel-Whitney Classes over Integers?
An interesting thing happened the other day. I was computing the Stiefel-Whitney numbers for $\mathbb{C}P^2$ connect sum $\mathbb{C}P^2$ to show that it was a boundary of another manifold. Of course, ...
- 2,684
23
votes
2
answers
2k
views
Why do people say DG-algebras behave badly in positive characteristic?
It seems to be a common wisdom in derived algebraic geometry that commutative DG-algebras are not, in general, a good model for derived rings, with the stated reason that they behave badly in positive ...
- 231
23
votes
1
answer
1k
views
A property of even continuous functions on the sphere
This question is inspired by On moments of inertia of planar and 3D convex bodies.
Let $f:{\mathbb R}^3\setminus\{0\}\to{\mathbb R}$ be an even homogeneous ($f(kx)=f(x)$ for all real $k\neq 0$) ...
- 91.8k
23
votes
5
answers
2k
views
Does anyone know a basepoint-free construction of universal covers?
Let $X$ be a real manifold (for simplicity). The standard construction of the universal cover $\varphi: \widetilde{X} \longrightarrow X$ involves fixing a basepoint $p \in X$ and considering homotopy ...
- 4,164
23
votes
4
answers
5k
views
Is $\ x\! \cdot\!\tan(x)\ $ integrable in elementary functions?
I'm teaching Calculus and my students asked me to calculate the integral of $\ x\! \cdot\!\tan(x)$.
I spent quite a lot of effort to do this, but I'm now even not sure if the integral could be ...
- 1,437
23
votes
4
answers
5k
views
De Rham decomposition theorem, generalisations and good references
De Rham decomposition theorem states that every simply-connected Riemannian manifold $M$
that admits complementary sub-bundles $T'(M)$ and $T''(M)$ of its tangent bundle parallel with respect to the ...
- 28.9k
23
votes
2
answers
6k
views
Definition of fiber bundle in algebraic geometry
If we have a map p: X --> Y of topological spaces, we can make a definition expressing that the topological type of the fibers of p varies continuously (edit: better to say "locally constantly", ...
- 9,283
23
votes
3
answers
1k
views
Is it possible to construct an action of an $E_\infty$ operad on $BU$ that respects filtration by $BU(n)$?
It is well known that $BU$ is an infinite loop space, and as such it has an action of an $E_\infty$ operad. An explicit construction of such an action is given, for example, in an answer to this MO ...
- 10.9k
23
votes
5
answers
2k
views
The "right" topological spaces
The following quote is found in the (~1969) book of Saunders MacLane,
"Categories for the working mathematician"
"All told, this suggests that in Top we have been studying
the wrong mathematical ...
- 18.7k
23
votes
4
answers
3k
views
What are Picard categories, where can I learn more about them, and why should I care to?
I have the category-theoretic background of the occasional stroll through MacLane's text, so excuse my ignorance in this regard. I was trying to learn all that I could on the subject of tensor ...
- 1,539
23
votes
3
answers
3k
views
Homology theory constructed in a homotopy-invariant way
Singular homology sends homotopic morphisms on equal morphisms and weakly equivalent spaces on isomorphic objects. So singular homology is in fact defined on the homotopy category of topological ...
- 3,033
22
votes
2
answers
2k
views
Non standard Algebraic Topology
Let *$\mathbb R$ a field of non-standard real numbers (or any real closed field) equipped with its natural generalized metric $d(x,y)=|x-y|$. Equip *$\mathbb R^2$ and *$\mathbb R^3$ with the $\ell^1$-(...
- 6,034
22
votes
0
answers
3k
views
Origins of the Nerve Theorem
Recently, I've read two papers which have cited the Nerve Theorem, one crediting Borsuk with the result and another Leray. Here is the question:
Who was the first to prove the Nerve Theorem?
- 15.5k
22
votes
0
answers
2k
views
Is the equivariant cohomology an equivariant cohomology?
Suppose a finite group $G$ acts piecewise linearly on a polyhedron $X$. Then there are two kinds of equivariant cohomology (or homology).
$\bullet$ With coefficients in a $\Bbb Z G$-module $M$. A ...
- 5,575
22
votes
4
answers
3k
views
Betti numbers of moduli spaces of smooth Riemann surfaces
Where can I find a list of the known Betti numbers of the moduli spaces $\mathcal{M}_{g,n}$ of genus $g$ Riemann surfaces with $n$ marked points? I need it to cross check results by an implemented ...
- 6,777
22
votes
4
answers
2k
views
Functorial Whitehead Tower?
The Whitehead tower of a (pointed) space is a tower of spaces which successively kills the bottom homotopy groups. The first two spaces can be constructed functorially (at least for suitably nice ...
- 27.5k
22
votes
5
answers
7k
views
Describing the universal covering map for the twice punctured complex plane
As is well known, the universal covering space of the punctured complex plane is the complex plane itself, and the cover is given by the exponential map.
In a sense, this shows that the logarithm has ...
- 5,530
22
votes
2
answers
6k
views
References and resources for (learning) chromatic homotopy theory and related areas
What references and resources (e.g. video recorded lectures) are available for learning chromatic homotopy theory and related areas (such as formal geometry)?
22
votes
2
answers
6k
views
Grothendieck's Tohoku Paper and Combinatorial Topology
I've read some discussions of Grothendieck's famous Tohoku Paper, and I understand that one reason it was a landmark paper was that it introduced abelian categories and gave us sheaf cohomology as a ...
- 221
22
votes
6
answers
2k
views
Is any interesting question about a group G decidable from a presentation of G?
We say that a group G is in the class Fq if there is a CW-complex which is a BG (that is, which has fundamental group G and contractible universal cover) and which has finite q-skeleton. Thus F0 ...
- 1,219
21
votes
2
answers
2k
views
Algebraic K-theory of the group ring of the fundamental group
I know of two places where $K_{*}(\mathbb{Z}\pi_{1}(X))$ (the algebraic $K$-theory of the group ring of the fundamental group) makes an appearance in algebraic topology.
The first is the Wall ...
- 726
21
votes
1
answer
2k
views
A spectral sequence for computing cohomology of a space from that of its strata
Let $X$ be a smooth complex variety (not necessarily compact) and let $D$ be a normal crossings divisors with components $D_1$, $D_2$, ..., $D_N$. For a set of indices $I$, let $D_I = \bigcap_{i \in I}...
- 156k
21
votes
0
answers
865
views
Does the Witten genus determine $\mathrm{tmf}$ (or $\mathrm{TMF}$)?
$\newcommand\specfont[1]{\mathrm{#1}}$$\newcommand\MSpin{\specfont{MSpin}}\newcommand\KO{\specfont{KO}}\newcommand\KU{\specfont{KU}}\newcommand\MString{\specfont{MString}}\newcommand\tmf{\specfont{tmf}...
- 6,777
21
votes
1
answer
983
views
Is the Alexander horned sphere a cofibration?
The Alexander horned sphere is a closed embedding of $S^2$ into $S^3$ which is not flat because otherwise the Schoenflies Theorem would be true for it. However, not being flat is not the same as not ...
- 263
21
votes
7
answers
1k
views
Reference for topological graph theory (research / problem-oriented)
I would be interested in recommendations for topological graph theory texts. I think Gross and Yellen has a great chapter on topological graph theory, and I find Mohar and Thomassen's Graphs on ...
21
votes
2
answers
3k
views
What is the 31st homotopy group of the 2-sphere?
What is $\pi_{31}(S^2)$, the 31st homotopy group of the 2-sphere ?
This question has a physics motivation:
There are relations between (2nd and 3rd) Hopf fibrations and (2- and 3-) qubits (quantum ...
- 275
21
votes
6
answers
3k
views
A ring such that all projectives are stably free but not all projectives are free?
This question is motivated by this recent question. Suppose $R$ is commutative, Noetherian ring and $M$ a finitely generated $R$-module. Let $FD(M)$ and $PD(M)$ be the shortest length of free and ...
- 30.6k
21
votes
2
answers
6k
views
Group completion theorem
Let $M$ be a topological monoid. How does the homology-formulation of the group completion theorem, namely (see McDuff, Segal: Homology Fibrations and the "Group-Completion" Theorem)
If $\pi_0$ is ...
- 1,085