All Questions
1,239 questions
22
votes
6
answers
3k
views
Does every vector bundle allow a finite trivialization cover?
Suppose there is a vector bundle (smooth, with constant rank finite-dimensional fibres) over a (smooth, second-countable, Hausdorff, not necessarily connected) manifold $B$ of dimension $n$.
(a) Is ...
- 1,284
21
votes
2
answers
4k
views
Topological $n$-manifolds have the homotopy type of $n$-dimensional CW-complexes
I search for a chain of clean references, which lead the fact of topological manifolds of dimension $n$ having the homotopy type of a CW of dimension $n$.
Milnor's On spaces having the homotopy type ...
- 489
21
votes
10
answers
6k
views
Not especially famous, long-open problems which higher mathematics beginners can understand
This is a pair to
Not especially famous, long-open problems which anyone can understand
So this time I'm asking for open questions so easy to state for students of subjects such as undergraduate ...
21
votes
5
answers
7k
views
Maps inducing zero on homotopy groups but are not null-homotopic
Today my fellow grad student asked me a question, given a map f from X to Y, assume $f_*(\pi_i(X))=0$ in Y, when is f null-homotopic?
I search the literature a little bit, D.W.Kahn
Link
And M....
- 1,160
21
votes
7
answers
3k
views
What should be taught in a 1st course on Riemann Surfaces?
I am teaching a topics course on Riemann Surfaces/Algebraic Curves next term. The course is aimed at 1st and 2nd year US graduate students who have have taken basic coursework in algebra and manifold ...
- 3,284
21
votes
3
answers
2k
views
Why is Kan's $Ex^\infty$ functor useful?
I've always heard that Kan's $Ex^\infty$ functor has important theoretical applications, but the only one I know is to show that the Kan-Quillen model structure is right proper. What else is it useful ...
- 64k
21
votes
1
answer
8k
views
When is a quasi-isomorphism necessarily a homotopy equivalence?
Under what circumstances is a quasi-isomorphism between two complexes necessarily a homotopy equivalence? For instance, this is true for chain complexes over a field (which are all homotopy ...
- 10.1k
21
votes
2
answers
2k
views
Topologically contractible algebraic varieties
From a post to The Jouanolou trick:
Are all topologically trivial (contractible) complex algebraic varieties necessarily affine? Are there examples of those not birationally equivalent to an affine ...
- 15.1k
21
votes
2
answers
2k
views
When is a topological space the homotopy colimit of an open covering?
Suppose that $X$ is a topological space and $\left(U_i \to X\right)$ is an open cover. We can associate to it the Cech diagram of this cover $$C_U:\Delta^{op} \to Top.$$ I know that for many good ...
- 15.5k
20
votes
2
answers
1k
views
The first unstable homotopy group of $Sp(n)$
Thanks to the fibrations
\begin{align*}
SO(n) \to SO(n+1) &\to S^n\\
SU(n) \to SU(n+1) &\to S^{2n+1}\\
Sp(n) \to Sp(n+1) &\to S^{4n+3}
\end{align*}
we know that
\begin{align*}
\pi_i(SO(...
- 19.3k
20
votes
3
answers
1k
views
Simultaneous "orthonormalization" in $\mathbb{C}^4$
Let $A$ be a positive, invertible $4 \times 4$ hermitian complex matrix.
So we have a positive sesquilinear form $\langle Av,w\rangle$. Say that a pair $(v,w)$ of vectors in $\mathbb{C}^4$ is good ...
- 42.8k
20
votes
4
answers
3k
views
Relationship between the cohomology of a group and the cohomology of its associated Lie algebra
Let $G$ be a group and let $k$ be a field (characteristic 0 if you want). Let $L$ be the graded Lie ring associated to the lower central series of $G$, that is, $L$, as a graded abelian group is $\...
- 373
20
votes
1
answer
2k
views
The cell structure of Thom spectra
I would like to understand the cell structure of integrally oriented Thom spectra. A Thom spectrum over a space $X$ is something you can build from a stable spherical bundle, which is classified by a ...
- 6,308
20
votes
3
answers
715
views
Which elements of $H^2(M,\mathbb{Z}/2)$ are the $w_2(E)$ for a real bundle $E$?
Any element of $H^1(M,\mathbb{Z}/2)$ is the $w_1(E)$ of a real line bundle $E$ over $M$.
I wonder how to characterize (probably using the Steenrod squares) which elements of $H^2(M,\mathbb{Z}/2)$ are ...
- 6,133
20
votes
3
answers
2k
views
Non-stably trivial bundle with trivial characteristic classes
Though it's relatively clear that the characteristic classes do not characterise a vector bundle (and after looking through some books) I could not find an example of a vector bundle which is not ...
- 4,432
19
votes
4
answers
2k
views
Details for the action of the braid group B_3 on modular forms
I'm reading Terry Gannon's Moonshine Beyond the Monster, and in section 2.4.3 he hints at (but does not explicitly describe) a way to extend the action of $SL_2(\mathbb{Z})$ on modular forms to an ...
- 118k
19
votes
3
answers
3k
views
When are (finite) simplicial complexes (smooth) manifolds?
Hi,
is there an algorithm that determines if a given simplicial complex is
a.) a manifold
b.) a smooth manifold
c.) homotopy equivalent to a manifold
d.) a real algebraic variety
?
- 949
19
votes
1
answer
2k
views
When does the free loop space fibration split?
This question is a repost from stack.exchange. It didn't get a lot of attention there. Perhaps it is badly written (or silly?). If so, I'd be happy to get comments/suggestions about that.
Let $X$ be ...
- 2,320
19
votes
4
answers
3k
views
What are the fibrant objects in the injective model structure?
If C is a small category, we can consider the category of simplicial presheaves on C. This is a model category in two natural ways which are compatible with the usual model structure on simplicial ...
- 27.5k
19
votes
1
answer
2k
views
Homotopy fiber of a map between classifying spaces
I'm looking for a reference (and precise hypothesis if more are needed) for the following facts (or a correction, if I'm just plain wrong):
Let $G$ and $H$ be topological groups and $f : G \to H$ be a ...
- 5,309
19
votes
14
answers
4k
views
Excellent uses of induction and recursion
Can you make an example of a great proof by induction or construction by recursion?
Given that you already have your own idea of what "great" means, here it can also be taken to mean that the chosen ...
19
votes
3
answers
2k
views
Is every paracompact, Hausdorff, locally contractible space homotopy equivalent to a CW complex?
Milnor proved that any paracompact Hausdorff space which is equi-locally convex (and hence in particular locally contractible) is homotopy equivalent to a CW complex. However, unlike being paracompact ...
- 9,481
19
votes
5
answers
2k
views
How do you define the strict infinity groupoids in Homotopy Type Theory?
In the setting of Homotopy Type Theory, how would you construct $\mathrm{isStrict} : U \rightarrow U$ which is inhabited exactly when the first type is (equivalent to?) a strict $\infty$-groupoid?
...
- 28.1k
18
votes
2
answers
1k
views
Is homology finitely generated as an algebra?
If a differential graded algebra is finitely generated as an algebra, is its homology finitely generated as an algebra?
Is it easier if we impose any of the three conditions: characteristic zero; ...
- 8,727
18
votes
3
answers
2k
views
Can eta invariant be written in terms of topological data?
The eta invariant was introduced by Atiyah, Patodi, and Singer. It roughly measures the asymmetry of the spectrum of a self-adjoint elliptic operator with respect to the origin. In the paper "Exotic ...
- 875
18
votes
3
answers
2k
views
Are finite spaces a model for finite CW-complexes?
Are finite topological spaces (i.e. topological spaces whose underlying set is finite) a model for the homotopy theory of finite simplicial sets (= homotopy theory of finite CW-complexes) ?
Namely, ...
- 43.2k
18
votes
2
answers
2k
views
Homotopy types of schemes
Let $X$ be a scheme over $\mathbb{C}$.
When does the topological space $X\left(\mathbb{C}\right)$ of $\mathbb{C}$-points have the homotopy type of a finite CW-complex?
When does the topological ...
- 15.5k
18
votes
1
answer
1k
views
Simply connected finite CW-complex with only finitely many nontrivial homotopy and homology groups
Let $X$ be a simply connected finite CW-complex such that all but finitely many of its homotopy groups and its homology groups (with $\mathbb Z$ coefficients) are 0.
Is $X$ then necessarily ...
- 11.9k
17
votes
8
answers
3k
views
Smooth classifying spaces?
Take G to be a group. I care about discrete groups, but the answer in general would be welcome too. There are the various ways to construct the classifying space of G, bar construction, cellular ...
- 1,422
17
votes
3
answers
1k
views
Codimension zero immersions
Given an immersion of the n-1-sphere into a (closed) n-manifold, when does it extend to an immersion of the n-disk?
Remark: If the sphere had dimension k smaller than n-1, then such an immersion ...
- 10.4k
17
votes
3
answers
4k
views
What is π_1(BG) for an arbitrary topological group $G$?
The classifying space $BG=|Nerve(G)|$ of an arbitrary topological group $G$ does not necessarily have the homotopy type of a CW-complex but the fundamental group should still be accessible. What is $\...
- 7,193
17
votes
2
answers
2k
views
Equivariant cohomology vs. invariant cohomology vs. cohomology of quotient space
Given a space $X$ and an action of a group $G$ on $X$, the $G$-invariant cochains with coefficients in an Abelian group $A$ define a sub-cocomplex $\mathcal{C}^{\bullet}_G$ of the cocomplex $\mathcal{...
- 863
17
votes
3
answers
2k
views
Is there Domain Invariance for Alexandrov spaces?
A colleague asked me this question recently. Every injective continuous map between manifolds of the same (finite) dimension is open - this is Brouwer's Domain Invariance Theorem. Is the same true for ...
- 32.4k
16
votes
4
answers
1k
views
Growth of stable homotopy groups of spheres
Let ${}_2\pi_n^S$ denote the $2$-power torsion subgroup of $n$th stable homotopy group of the sphere spectrum. Its order is a power of $2$: $$|{}_2\pi_n^S|=2^{k_n}.$$
Question: What is known about ...
- 1,484
16
votes
1
answer
2k
views
Why does the singular simplicial space geometrically realize to the original space?
I have seen it claimed that (for compactly generated Hausdorff spaces) the geometric realization of the singular (internal) simplicial space is homotopy equivalent to the original space. I know how to ...
- 27.5k
16
votes
1
answer
1k
views
Easiest example where pseudo-isotopy fails to be the same as isotopy?
This question concerns diffeomorphism of manifolds. Let $f: M \to M$ be a self-diffeomorphism. We will say that it is isotopic to the identity if there is a continuous one-parameter family of ...
- 27.5k
16
votes
3
answers
3k
views
Multiplicativity of Euler characteristic for non-orientable fibrations
Let $E\to B$ be a fibration with fiber F, and assume for simplicity that B is connected. Suppose moreover that B and F have Euler characteristics (perhaps they are manifolds). Then often, one can ...
- 66.8k
16
votes
3
answers
4k
views
Homotopy Groups of Connected Sums
This was sparked because I wanted to compute $\pi_2(Sym^2(\Sigma_2))$ via $Sym^2(\Sigma_2)\approx \mathbb{T}^4$# $\bar{\mathbb{C}P}^2$.
We know how to compute $\pi_1$ of $M$ # $N$ via van-Kampen's ...
- 17.5k
16
votes
4
answers
2k
views
Coboundaries and Gluing in Cech Cohomology - Intuition?
I'm trying to develop an intuition for Cech cohomology geometrically, but am currently failing. A lot of people seem to say that the groups $H^n$ measure obstructions to gluing local sections to get ...
- 827
16
votes
10
answers
3k
views
Orbifold fundamental group in terms of loops?
In chapter 13 in his notes on 3-manifolds, Thurston defines the orbifold fundamental group to be the group of deck transformations of the universal cover of the orbifold. He also makes a statement "...
- 13.6k
16
votes
2
answers
822
views
Klee's trick --- more applications
In his "Some topological properties..." (1955), Klee gave a construction (simple and beautiful) of an isotopy $h_t\colon\mathbb{R}^{2\cdot n}\to \mathbb{R}^{2\cdot n}$ which moves any compact set $K$ ...
- 45k
16
votes
2
answers
878
views
Spaces with both "simple homology" and "simple homotopy" at the same time
Maybe every algebraic topology student, at some moment, will ask himself/herself the question: why are $\pi_*$ so difficult and mysterious, especially when compared with (co)homology? Think about the ...
- 1,525
15
votes
3
answers
1k
views
Strictly commutative elements of $E_\infty$-spaces
Let $X$ be an $E_\infty$-space (not necessarily grouplike). Let $x \in \pi_0 X$ be an element; say that $x$ is strictly commutative if there is a map of $E_\infty$-spaces $\mathbb{Z}_{\geq 0} \to X$ ...
- 25.6k
15
votes
2
answers
1k
views
"Economic" CW-structure for Eilenberg-MacLane spaces?
The only really "economic" cell structures for $K(\pi,n)$'s that I know is the one with a single cell in each dimension for $K(\mathbb Z/n\mathbb Z,1)$ and the one with a single cell in each even ...
- 18.4k
15
votes
2
answers
1k
views
"Strøm-type" model structure on chain complexes?
Background
The Quillen model structure on spaces has weak equivalences given by the weak homotopy equivalences and the fibrations are the Serre fibrations. The cofibrations are characterized by ...
- 18.9k
15
votes
2
answers
2k
views
Every 4-manifold has a $\operatorname{Spin}^c$ Structure
$\DeclareMathOperator\Spin{Spin}\DeclareMathOperator\SO{SO}$I'm having trouble understanding the proof given in Morgan's The Seiberg–Witten Equations and Applications to the Topology of Smooth Four-...
- 151
15
votes
5
answers
3k
views
Generalization of winding number to higher dimensions
Is there a natural geometric generalization of the winding number to higher dimensions?
I know it primarily as an important and useful index for closed, plane curves
(e.g., the Jordan Curve Theorem),
...
- 151k
15
votes
1
answer
513
views
fundamental groups of complements to countable subsets of the plane
This question is a follow-up of this MSE post and a comment by Henno Brandsma:
Question 1. Let $S$ be the set of isomorphism classes of fundamental groups $\pi_1(E^2 - C)$, where $C$ ranges over all ...
- 12.3k
14
votes
1
answer
981
views
Characteristic classes for odd $K$-theory
There are different models of odd $K$-theory. In one case,
one takes the group $U=\lim\limits_{\longrightarrow}U(n)$ as classifying space. Similarly, if $\mathcal U$ denotes the unitary group of a ...
- 6,813
14
votes
2
answers
2k
views
Well-pointed space which is not locally contractible
I am looking for an example of a well-pointed space in which no (sufficiently small) neighbourhood of the base-point is contractible. As usual, a well-pointed space is a pointed space in which the ...
- 6,210