All Questions
9,056 questions
29
votes
3
answers
2k
views
Categorification of determinant
The notion of trace of a matrix can be generalized to trace of an endomorphism of a dualizable objects in a symmetric monoidal category. (See Ponto & Shulman for a nice description.)
Is there a ...
- 290
29
votes
4
answers
1k
views
Which stable homotopy groups are represented by parallelizable manifolds?
The Pontryagin-Thom construction allows one to identify the stable homotopy groups of spheres with bordism classes of stably normally framed manifolds. A stable framing of the stable normal bundle ...
- 27.5k
29
votes
4
answers
3k
views
Geometric interpretation of the lower central series for the fundamental group?
For any group $G$ we can form the lower central series of normal subgroups by taking $G_0 = G$, $G_1 = [G,G]$, $G_{i+1} = [G,G_i]$. We can check this gives a normal chain
$$G_0 \ge G_1 \ge ... \ge G_i ...
- 573
29
votes
3
answers
2k
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Is the moduli space of unorientable Riemann surfaces with $pin^+$ structure orientable?
By a non-orientable Riemann surface ${\cal C}$, I mean a compact non-orientable two-manifold without boundary that is endowed with a conformal structure.
Such objects have a moduli space that is ...
- 1,369
29
votes
5
answers
3k
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Topologists loops versus algebraists loops
Let X be an affine variety over ℂ. Consider X(ℂ) with the classical topology, and create the topologists loop space ΩX(ℂ) of maps from the circle into X(ℂ). One can also ...
- 8,866
29
votes
2
answers
1k
views
Quillen + construction for finite groups
Is there an example of two non isomorphic finite groups $G$ and $H$ such that $BG^{+}$ is homotopy equivalent to $BH^{+}$ ?
- 1,629
29
votes
2
answers
2k
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Equivariant classifying spaces from classifying spaces
Given compact Lie groups $G$ and $\Pi$, there is a notion of "$G$-equivariant principal $\Pi$-bundle", and a corresponding notion of classifying space, often denoted $B_G\Pi$, so that $G$-equivariant ...
- 27.2k
29
votes
1
answer
1k
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Is the Golomb countable connected space topologically rigid?
The Golomb space $\mathbb G$ is the set of positive integers endowed with the topology generated by the base consisting of the arithmetic progressions $a+b\mathbb N_0$ with relatively prime $a,b$ and $...
- 41.9k
29
votes
1
answer
2k
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Does the Gauss-Bonnet theorem apply to non-orientable surfaces?
I hesitated for a long time to ask such an elementary-seeming question on Math Overflow, but when I asked and bountied it on Math SE, I found that a few experts seem to disagree on the answer, and I ...
- 1,311
29
votes
1
answer
2k
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High-Dimensional Analogs of Polygon Spaces
[Edit: I had a mistake in the numerology (took d=6,5 instead of d=5,4). Edit: I mistakenly identified my mistake, it is 6,5 but I got the indices shifted by one.]
Background: Polygon spaces
Given a ...
- 24.7k
29
votes
1
answer
1k
views
Software for rational homotopy theory
Does anybody know a software manipulating commutative differential graded algebras, and providing a computation of the minimal model? I tried to use the package DGAlgebras of Macaulay2, but I got ...
- 466
29
votes
0
answers
3k
views
Why do polytopes pop up in Lagrange inversion?
I'd be interested in hearing people's viewpoints on this. Looking for an intuitive perspective. See Wikipedia for descriptions of polytopes and the Lagrange inversion theorem/formula (LIF) for ...
28
votes
5
answers
3k
views
Is there a Morse theory proof of the Bruhat decomposition?
Let $G$ be a complex connected Lie group, $B$ a Borel subgroup and $W$ the Weyl group. The Bruhat decomposition allows us to write $G$ as a union $\bigcup_{w \in W} BwB$ of cells given by double ...
- 8,167
28
votes
2
answers
6k
views
What group is $\langle a,b \,| \, a^2=b^2 \rangle$?
In teaching my algebraic topology class, this group showed up as part of an easy fundamental group computation: $\langle a,b\mid a^2=b^2\rangle$. My first instinct was that this must be $\mathbb{Z}*\...
- 5,534
28
votes
5
answers
5k
views
Are rational varieties simply connected?
Is it true that every smooth rational variety X is simply connected? How is the proof?
Would it be still true if X has mild (for example orbifold) singularities?
28
votes
5
answers
9k
views
Why can't the Klein bottle embed in $\mathbb{R}^3$?
Using Alexander duality, you can show that the Klein bottle does not embed in $\mathbb{R}^3$. (See for example Hatcher's book Chapter 3 page 256.) Is there a more elementary proof, that say could be ...
- 1,355
28
votes
6
answers
4k
views
How should I visualise RP^n?
So I did some algebraic topology at university, including homotopy theory and basic simplicial homology, as well as some differential geometry; and now I'm coming back to the subject for fun via ...
- 1,180
28
votes
1
answer
2k
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Example of 4-manifold with $\pi_1=\mathbb Q$
This might be well known for algebraic topologist. So I am looking for an explicit example of a 4 dimensional manifold with fundamental group isomorphic to the rationals $\mathbb Q$.
- 2,623
28
votes
2
answers
2k
views
Why study the p-completions of a space?
Given a nice topological space $X$ there are various notions of a 'completion' at a set of primes. Some of the most common constructions may be found in Bousfield-Kan's, May's, Neisendorfer's or ...
- 5,596
28
votes
5
answers
4k
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Modern source for spectra (including ring spectra)
I am looking for a modern introduction to spectra that improves on the treatment by Adams in his "Stable Homotopy and Generalized Homology" notes (by improves I mean taking into account what ...
28
votes
3
answers
2k
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Does a *topological* manifold have an exhaustion by compact submanifolds with boundary?
If $M$ is a connected smooth manifold, then it is easy to show that there is a sequence of connected compact smooth submanifolds with boundary $M_1\subseteq M_2\subseteq\cdots$ such that $M=\bigcup_{i=...
- 18.7k
28
votes
3
answers
2k
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A non-formal space with vanishing Massey products?
Let $X$ be a polyhedron. For each $n$-dimensional face $f$ of $X$ fix a homeomorphism $\sigma_f:\triangle^n\to f$ where $\triangle^n$ is the standard $n-$simplex so that whenever $f$ is a face of $f'$ ...
- 23.5k
28
votes
3
answers
1k
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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
28
votes
4
answers
4k
views
(∞, 1)-categorical description of equivariant homotopy theory
I'm trying to learn a bit about equivariant homotopy theory. Let G be a compact Lie group. I guess there is a cofibrantly generated model category whose objects are (compactly generated weak ...
- 25.2k
28
votes
2
answers
2k
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Has anyone seen a nice map of multiplicative cohomology theories?
I have seen lots of descriptions of this map in the literature but never seen it nicely drawn anywhere.
I could try to do it myself but I really lack expertise, hence am afraid to miss something or ...
- 18.4k
28
votes
5
answers
4k
views
Two-to-one continuous mapping from R² to R²
Hello. I have a question.
Does there exist a continuous mapping
$F:\mathbb{R}^2\rightarrow\mathbb{R}^2$
such that for every $c\in F(\mathbb{R}^2)$
there are two and only two points $z_{1}$, $z_{2}$...
- 301
28
votes
2
answers
1k
views
Is there a geometric realization of $\mathbf{C}((t))$-varieties?
Let $MV_F$ be the $\mathbf{A}^1$-homotopy category over the field $F$. Let $H$ be the homotopy category of spaces, and let $H_{/S^1}$ be the homotopy category of spaces over the circle.
When $F = \...
- 4,949
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\ .
$$
I want to understand the classifying space of $G$. Since ...
- 4,217
28
votes
1
answer
2k
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In M-theory, what can hypothesis H tell us that quantization in ordinary cohomology cannot?
In classical field theory, many fields and related objects are described as differential
forms. For example, in electromagnetism, the field $F := B - \mathrm dt\wedge E$ is a 2-form, and Maxwell's
...
- 6,881
28
votes
1
answer
3k
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Two points of view about Borel-moore homology
They are several ways to define the Borel-Moore homology on a locally compact space $X$.
The first one is by analogy with the singular homology but instead of using finite chains, we use locally ...
- 1,017
28
votes
1
answer
835
views
Modern survey of unstable homotopy groups?
Toda no doubt made some big strides when computing unstable homotopy groups $\pi_{n+k}(S^n)$ for $k < 20$ which his collaborators later improved upon.
The methods he used are documented in his ...
- 289
28
votes
1
answer
1k
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Is there a general theory of fiber theorems?
Here are three vague theorems rolled up in one.
Let $X$ and $Y$ be sufficiently nice topological spaces and $f:X \to Y$ a sufficiently nice surjection. If for each $y \in Y$, the fiber $f^{-1}(y) \...
- 15.5k
28
votes
1
answer
856
views
Is there a Kan-Thurston theorem for fibrations ?
Given a fibration $F \to X \to B$ with all spaces path-connected. Is there a (discrete) group $G$ with normal subgroup $H$ such that
$$H^\ast(BG;\mathcal{A}) = H^\ast(X;\mathcal{A})$$
$$H^\ast(BH;\...
- 16.2k
28
votes
0
answers
1k
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On the (derived) dual to the James construction.
Background
If $X$ is a based space then the James construction on $X$ is the space $J(X)$ given by
$$
X \quad \cup \quad X^{\times 2} \quad \cup \quad X^{\times 3} \quad \cup \quad \cdots
$$
in ...
27
votes
3
answers
6k
views
Why is BG infinite dimensional for G finite ?
If $G \neq \lbrace 1 \rbrace$ is a finite group with classifying space $BG$
then there are infinitely many i such that $H^i(BG,\mathbb{Z}) \neq 0$. This
can be found, for example, there:
Non-...
- 2,160
27
votes
5
answers
5k
views
What is the intuition behind the Freudenthal suspension theorem?
The Freudenthal suspension theorem states in particular that the map
$$
\pi_{n+k}(S^n)\to\pi_{n+k+1}(S^{n+1})
$$
is an isomorphism for $n\geq k+2$.
My question is: What is the intuition behind the ...
- 727
27
votes
8
answers
3k
views
Object of proven finiteness, yet with no algorithm discovered?
I explain my title by two examples in number theory:
The rational points on elliptic curve over number fields forms a finitely generated abelian group, so its rank is an integer, but so far we do not ...
- 1,053
27
votes
6
answers
4k
views
Failure of smoothing theory for topological 4-manifolds
Smoothing theory fails for topological 4-manifolds, in that a smooth structure on a topological 4-manifold $M$ is not equivalent to a vector bundle structure on the tangent microbundle of $M$. Is ...
- 967
27
votes
13
answers
4k
views
Homological algebra for commutative monoids?
Homological algebra for abelian groups is a standard tool in many fields of mathematics. How much carries over to the setting of commutative monoids (with unit)? It seems like there is a notion of ...
- 27.5k
27
votes
2
answers
4k
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What's the current state of the classification of not-fully-extended TQFTs?
Recall that a $(k,k+1,\dots,k+n)$-TQFT is (supposed to be) a functor from the $n$-category whose $j$-morphisms are (isomorphism classes of) compact $(k+j)$-dimensional manifolds with boundary to some ...
- 54.6k
27
votes
3
answers
4k
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"Dirty" proof that Eilenberg-MacLane spaces represent cohomology?
The standard approach to proving that $H^n(X; G)$ is represented by $K(G, n)$ seems to be to prove that $\text{Hom}(X, K(G, n))$ defines a cohomology theory and then use Eilenberg-Steenrod uniqueness. ...
- 2,168
27
votes
6
answers
3k
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Applications of string topology structure
Chas and Sullivan constructed in 1999 a Batalin-Vilkovisky algebra structure on the shifted homology of the loop space of a manifold: $\mathbb{H}_*(LM) := H_{*+d}(LM;\mathbb{Q})$. This structure ...
- 8,167
27
votes
3
answers
7k
views
Why are we interested in the Fundamental Groupoid of a Space?
The classical version of the van Kampen theorem is concerned about the fundamental group of a based space. In fact, it says that the functor $\pi_1$ preserves certain types of pushouts in $Top_*$.
...
- 609
27
votes
3
answers
2k
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Fundamental group of a topological pullback
This should be such an elementary problem in algebraic topology that I'm almost too embarrassed to ask, but here goes.
Let $f: X\to Z$ be a surjective fibration, and let $g: Y\to Z$ be any map. ...
- 35.9k
27
votes
2
answers
2k
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Atiyah-Singer style index theorem for elliptic cohomology?
In 1994, Mike Hopkins wrote a paper called Topological Modular Forms, the Witten Genus, and the Theorem of the Cube. As usual, the introduction was fantastic, explaining the power of various cobordism ...
- 30.3k
27
votes
6
answers
4k
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A toolbox for algebraic topology
This question has a very general part and a rather concrete part.
General:
When one wants to prove something in algebraic topology (actually in all parts of mathematics) one obviously needs some ...
27
votes
3
answers
2k
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Configuration space of little disks inside a big disk
The space of configurations of $k$ distinct points in the plane
$$F(\mathbb{R}^2,k)=\lbrace(z_1,\ldots , z_k)\mid z_i\in \mathbb{R}^2, i\neq j\implies z_i\neq z_j\rbrace$$
is a well-studied object ...
- 35.9k
27
votes
5
answers
9k
views
Textbook or lecture notes in topological K-Theory
I am looking for a good introductory level textbook (or set of lecture notes) on classical topological K-Theory that would be suitable for a one-semester graduate course. Ideally, it would require ...
27
votes
3
answers
6k
views
The relationship between group cohomology and topological cohomology theories
I was recently trying to learn a little bit about group cohomology, but one point has been confusing me. According to wikipedia (https://en.wikipedia.org/wiki/Group_cohomology and some other sources ...
- 757
27
votes
2
answers
4k
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Why the sphere spectrum is more correct than $\mathbb{Z}$?
One may argue that $\mathbb{S}$ is more correct than $\mathbb{Z}$. Can anyone make it more explicitly? For example, what information will be lost if we work in $\mathbb{Z}$ instead of $\mathbb{S}$？
...
- 1,168