All Questions
9,056 questions
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 ...
8
votes
0
answers
172
views
The pro-discrete space of quasicomponents of a topological space
Let $X$ be a topological space.
Consider the functor $P^X : \textbf{Set} \to \textbf{Set}$ that sends each set $Y$ to the set of continuous maps $X \to Y$.
It is not hard to check that $P^X : \textbf{...
- 15.9k
6
votes
0
answers
748
views
How to learn homotopy theory
I studied some basic algebraic topology (homotopy/homology/cohomology groups). When reading about the Dold-Thom theorem, the fancier and more recent sources sooner or later all started to use homotopy ...
- 163
62
votes
9
answers
9k
views
Fundamental groups of noncompact surfaces
I got fantastic answers to my previous question (about modern references for the fact that surfaces can be triangulated), so I thought I'd ask a related question. A basic fact about surface topology ...
- 44.8k
17
votes
2
answers
725
views
For which $n$ does there exist a closed manifold of (chromatic) type $n$?
Let $p$ be a prime and $n \in \mathbb N$. Does there exist a closed manifold which is of type $n$ after $p$-localization?
When $n= 0$ the answer is yes. When $p = 2$ and $n = 1$ we can take $\mathbb R ...
- 64k
2
votes
0
answers
68
views
Sampling theorems for partition polynomials (associahedra, noncrossing partitions / parking functions)
Define the associahedra partition polynomial
$$
\begin{split}
A(x) &= 1 + A_1(u_1) z + A_2(u_1,u_2) z^2 + A_3(u_1,u_2,u_3) z^3 + \cdots\\
& \qquad\qquad = 1 + \sum_{n \geq 1} A_n(u_1,...,u_n) ...
- 10.5k
9
votes
1
answer
588
views
Oriented bordism in higher dimensions (e.g. $12 \leq d \leq 28$)
The classification of oriented compact smooth manifolds up to oriented cobordism is one
of the landmarks of 20th century topology. The techniques used there form the part of the foundations of ...
- 10.5k
2
votes
0
answers
66
views
Coker of map $\alpha_1(11)^*\oplus \alpha_2(11)^*$
How to calculate the 3-local Coker of map $\pi_{11}(Sp(3))\overset{{\alpha_1(11)}^* \oplus {\alpha_2(11)}^*}\longrightarrow \pi_{14}(Sp(3))\oplus \pi_{18}(Sp(3))$?
where $\pi_{11}(Sp(3))\cong \mathbb{...
7
votes
2
answers
619
views
Status of the Hopf-Thurston sign conjecture in dimension 4
A famous conjecture in topology asserts:
The Euler characteristic of a closed aspherical $2n$-manifold $M$ satisfies $(-1)^n\chi(M) \geq 0$.
This was conjectured by Hopf for manifolds with non-...
- 11.9k
5
votes
1
answer
273
views
Monoidal Dold–Kan correspondence for non-connected CDGA
Why can't the monoidal Dold–Kan correspondence be extended to non-connected CDGAs over a field of characteristic 0?
I understand that there is a technical problem with the original proof due to ...
- 1,374
0
votes
1
answer
205
views
Space-time trajectory that cannot be straightened and its braid form
Considering we have the space-time trajectory of multiple particles (or any objects) in the X-Y-Time coordination system. Given a projection direction, we can obtain the braid form of the space-time ...
- 73
1
vote
2
answers
482
views
(Lower) homotopy groups from triangulations
Both cohomology and homotopy groups capture global topological information of a manifold $X$. It is interesting to ask if they can be computed from local data. A triangulation $T$ is a natural ...
- 5,230
0
votes
0
answers
43
views
Intersection of subspace of cyclical rotations with orthant
In an $N$-dimensional real Euclidian space, let an orthant be specified by a vector
$\underline{x}_0 = \{x_1, x_2, \dots, x_N\}$ where the components $x_k$ are binary in the sense that $x_k = \pm 1$...
- 101
3
votes
1
answer
313
views
Homology of braid groups and loop spaces
How do Segal's theorems from (Configuration-spaces and iterated loop-spaces. Invent. Math.21:213--221)
imply that there is an isomorphism $H_*(B_\infty,\mathbb{Z})\cong H_*(\Omega^2S^3,\mathbb{Z})$,
...
- 191
1
vote
2
answers
243
views
coset poset of reflection subgroup
Fix a finitely generated Coxeter system $(W, S)$, and let $W_J$ denote the standard
parabolic proper subgroup generated by a subset $J \subset S$. It is well
known that the poset of cosets $\{xW_J\}$ ...
- 91
23
votes
2
answers
2k
views
Latest results in chromatic homotopy theory
I started a PhD in chromatic homotopy three years ago, but I had to quit it due to personal reasons after one year. Last week I was looking at all my notes from that period and I was wondering where ...
- 899
10
votes
2
answers
405
views
Multiplicative structures on truncated Moore spectra
As discussed for example in this paper (and this MO thread), there are obstructions for the existence of $A_n$-structures on Moore spectra $M(p^r):=\mathbb{S}/p^r$, which in general don't vanish. In ...
- 2,320
3
votes
1
answer
353
views
Are there strictly connective smooth proper algebras over $\mathbb{F}_p$?
Are there any connective $E_1$ rings $R$ over $\mathbb{F}_p$ satisfying the following?
$\pi_*(R)$ is a finite dimensional $\mathbb{F}_p$ vector space
$R$ is compact as a module over $R \otimes R^{op}...
- 2,356
2
votes
0
answers
60
views
Module structure of $\Omega_*(Z_p)$
In Conner-Floyd's book, Differentiable Periodic Maps in (46.1), for p an odd prime and $k=1,2,\dots$, it is posted the identities:
$$p\alpha_{2k+1}+[M^4]\alpha_{2k-3}+[M^8]\alpha_{2k-7}+\dots=0$$
in $\...
3
votes
1
answer
525
views
The notion of abelian covers
I have some doubts about what an abelian covering is, and I'll try my best to articulate them.
In Serre's Algebraic groups and class fields Chapter VI.2, he fixed a base field $k$ with algebraic ...
- 895
1
vote
0
answers
119
views
Equivariant cohomology with compact support of the generalized Jacobian of a nodal elliptic curve
[Question forwarded from SE for lack of interaction]
Given an geometric genus $1$ curve with $1$ node $C$ (hence with arithmetic genus $2$), its normalization is given by the elliptic curve $\tilde{C}$...
- 33
15
votes
1
answer
1k
views
Is cohomology always related to topology?
I am trying to understand whether there is a sense in which cohomology always relates to topology or whether this is the case only in particular examples. According to the Wikipedia page, a cochain ...
- 333
31
votes
9
answers
5k
views
Why should I prefer bundles to (surjective) submersions?
I hope this question isn't too open-ended for MO --- it's not my favorite type of question, but I do think there could be a good answer. I will happily CW the question if commenters want, but I also ...
- 54.6k
4
votes
1
answer
147
views
Intersection form of $2n$-manifold for odd $n$
Let $M$ be closed orientable $2n$-manifold, where $n$ is odd. It is well known that the $\mathbb Z$-module $H^\bullet(M;\mathbb Z)$ has graded-commutative multiplication and $H^{2n}(M;\mathbb Z)\simeq\...
- 1,095
1
vote
0
answers
131
views
Definition of Cartan Model - Equivariant differential forms
I would like to ask about an equivalence between two definitions for the Cartan Model.
Let $G$ be a connected Lie group, let $\mathfrak{g}$ be its Lie algebra and let $M$ be a $G$-manifold. The Cartan ...
- 11
4
votes
0
answers
373
views
Are Frobenius modules related to Frobenius algebras?
Frobenius modules appear in the Riemann Hilbert correspondence.
Frobenius algebras appear in TQFT.
Is there a way to pass from one to the other?
- 705
104
votes
10
answers
24k
views
Motivation for algebraic K-theory?
I'm looking for a big-picture treatment of algebraic K-theory and why it's important. I've seen various abstract definitions (Quillen's plus and Q constructions, some spectral constructions like ...
14
votes
1
answer
3k
views
Entering to the K-theory realm
I am looking for a guidance in $K$-theory. My master thesis was in the field of Algebraic K-theory and its relation
and interaction with the field of Algebraic Topology. I mainly had
concentrated on ...
- 309
2
votes
1
answer
369
views
$4$-manifold with simply connected boundary
This may be a very silly question but I could not get any counter-example.
Let $M$ be a compact differential $4$-manifold with boundary $dM$.
Suppose that the inclusion map induced map $\pi_1(dM) \to \...
- 1,177
63
votes
5
answers
18k
views
What is modern algebraic topology(homotopy theory) about?
At a basic level, algebraic topology is the study of topological spaces by means of algebraic invariants. The key word here is "topological spaces". (Basic) algebraic topology is very useful in other ...
11
votes
2
answers
1k
views
If two smooth manifolds are homeomorphic, then their stable tangent bundles are vector bundle isomorphic
I am currently reading Kervaire-Milnor's paper "Groups of Homotopy Spheres I", Annals of Mathematics, and I am trying to prove (or disprove) the following result. The more elementary the ...
- 881
3
votes
0
answers
79
views
Tautological ring for moduli of flat connections
Let $X$ be a smooth complex manifold and $G$ a connected complex algebraic group. Let $M$ denote the moduli stack of flat $G$-connections on $X$. Over $M\times X$, we have the tautological $G$-bundle, ...
- 2,751
21
votes
1
answer
1k
views
Does every group arise as the fundamental group of a complete Kähler manifold?
The fundamental group of a manifold is countable, and every countable group $G$ arises as the fundamental group of a (smooth) manifold; see this comment or this answer for a construction of an open ...
- 19.3k
7
votes
2
answers
576
views
Simplicial nerve of a topological group
Given a group $G$, one way to construct the classifying (topological) space $BG$ is to regard $G$ as a category with one object and morphisms $G$, take its nerve, and then apply the geometric ...
- 2,292
7
votes
1
answer
240
views
If the number of ends of Freudenthal space is infinite, then its space of ends is homeomorphic to the Cantor set?
I don't know whether this is the right place to discuss a part of someone's thesis or not. If it is wrong, let me know; I will delete my post.
I am reading this thesis.
Corollary 4.1.15. on page 63 ...
- 1,097
3
votes
1
answer
160
views
Definition of S-reducibility and reducibility of a space
I was going through this paper by Tanaka but I am stuck at Proposition 4.1 given below . I just cannot make sense of the first two lines of the proof. What does it mean when he says S-reducible and ...
3
votes
0
answers
179
views
Morse theory for isotopies of codimension 1 submanifolds and generalized bigon moves
I seem to have managed to convince myself of the validity of a certain result. If it does indeed hold (modulo non-catastrophic adjustments) I could really use a reference. Otherwise, I would also be ...
- 151
5
votes
0
answers
260
views
What is the winding behavior of the Riemann zeta function around zero along the line $s=1+it$?
Let $\phi: \mathbb R \setminus \{0\} \to S^1 \subset \mathbb C$ be defined by
$$\phi(t)= \zeta(1+it)/|\zeta(1+it)|$$
(the nonvavishing of the denominator being a bit weaker than the prime number ...
- 64k
2
votes
1
answer
564
views
Is there an operad homotopifying the Koszul rule?
In homotopy theory one has the idea of a homotopy-commutative multiplication, in which one replaces the relation $$ab=ba$$ in a commutative monoid/group/ring/etc. for an unspecified homotopy. One ...
- 11.8k
3
votes
0
answers
161
views
Making the powerset into a topological monoid
Every monoid $X$ induces a monoid structure $\circledast$ on $\mathcal{P}(X)$ via
$$U\circledast V := \{uv\ |\ u\in U,v\in V\}.$$
Moreover, a morphism of monoids $f\colon X\to Y$ induces a morphism of ...
- 11.8k
6
votes
0
answers
145
views
Mapping space between $n$-groupoids is an $n$-groupoid
Consider two simplicial sets $K$ and $L$. Their mapping space (or mapping complex) is the internal hom of simplicial sets, i.e. $\underline{\mathrm{Hom}}(K,L)$, where
$$
\underline{\mathrm{Hom}}(K,L)...
- 91
9
votes
2
answers
945
views
Stable homotopy groups of complex projective plane
We know that there is a cofiber sequence $S^3\xrightarrow{\eta}S^2\to\mathbb{C}\mathbb{P}^2$. It's easy to know that $\pi_3^s(\mathbb{C}\mathbb{P}^2)=0$ so there is a surjection
$$\partial:\pi_7^s(S^2\...
- 918
1
vote
0
answers
123
views
Relation of branched covers and groups
I am self-studying covering spaces of topological spaces. The following question comes to my mind.
In the case of topological covering spaces, we have a nice relation between the fundamental group of ...
- 613
6
votes
1
answer
632
views
Applications of $\mathbb{Z}$-graded algebraic geometry to algebraic topology
There's a theory of algebraic geometry over $\mathbb{Z}_2$-graded commutative rings, often called "algebraic supergeometry" or the theory of superschemes. From what I understand, there's ...
- 11.8k
0
votes
1
answer
477
views
Proving the induced map on the cohomology is an isomorphism
I was going through a paper by Tanaka where I am stuck at the following map "f" which is given by the composition of these maps. Next, he mentions that the induced map is clearly an ...
8
votes
1
answer
474
views
Is there a Dold-Kan theorem for circle actions?
There are several interesting equivalences of "Dold-Kan type" in the setting of stable $\infty$-categories. Namely, let $\mathcal C$ be a stable $\infty$-category. Then the following 3 ...
- 64k
25
votes
1
answer
2k
views
What can we say about the Cartesian product of a manifold with its exotic copy?
Let $M$ be a smooth oriented manifold, and let $M^E$ be an exotic copy, i.e homeomorphic but not diffeomorphic to $M$.
Is it true that $M\times M$ is diffeomorphic to $M\times M^E$?
I am ...
- 3,751
25
votes
3
answers
1k
views
What spaces $X$ do have $\text{End}(X) \cong \text{End}(\mathbb{R})$?
This is a follow-up on the following question. Let $\text{End}(X)$ denote the endomorphism monoid of a topological space $X$ (that is, the collection of all continuous maps $f:X\to X$ with composition)...
- 48.9k
8
votes
1
answer
399
views
Universal cover with one end
Suppose that $M$ is a non-compact manifold of finite topological type with one end which is the universal cover of some closed manifold $N$.
Is $M $ necessarily homeomorphic to the total space of some ...
- 6,995
2
votes
1
answer
226
views
Dualizing complex of the cone over a manifold
Let $M$ be a smooth (or just topological) closed manifold. Let $C(M)$ denote the cone over $M$, i.e.
$C(M)$ equals to $M\times [0,\infty)$ with $M\times \{0\}$ contracted to a point. The image of $M\...
- 21.8k