All Questions
9,056 questions
5
votes
1
answer
221
views
Serre spectral sequence of Borel construction
Let $G$ be finite $p$-group, $X$ be path-connected $G$-space, $E=EG\times_{G}X$ be the Borel construction and $BG$ be the classifying space of $G$. Consider Serre spectral sequence of the fibration
$$...
- 51
34
votes
6
answers
4k
views
Why study finite topological spaces?
In rereading Thurston's essay On Proof and Progress in Mathematics I ran across this passage:
… this means that some concepts that I use freely and naturally in
my personal thinking are foreign to ...
- 737
7
votes
1
answer
435
views
What are the covering spaces of $\mathbb{Q}$?
Let $X = \mathbb{Q}$, topologized as a subset of the real line. Is there a reasonable description of the covering spaces of $X$?
Here is something more precise. One way of constructing covers $p: \...
- 395
0
votes
0
answers
92
views
About filtration of the Leray-Serre spectral sequence
In the following proof, it is used the spectral sequence of the Borel
fibration $X\longrightarrow X_{T}\longrightarrow B_{T}$.
I don't understand how the map $\psi $ is obtained, and how is it an $R$-...
- 1,367
28
votes
3
answers
1k
views
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
1
vote
0
answers
125
views
Relating singular homology of function spaces: a natural transformation from $C(\mathbb{R}, -)$ to $L^p(\mathbb{R}, -)$
Consider the category $\mathcal{Top}_*$ of pointed topological spaces and continuous basepoint-preserving maps. Let $C(\mathbb{R}, X)$ denote the space of continuous maps from the real line $\mathbb{R}...
user538396
8
votes
0
answers
287
views
What is the current research situation of the Cheeger–Goresky–MacPherson conjecture?
In [CGM-1983], J. Cheeger, M. Goresky and R. MacPherson conjectured that the intersection cohomology of a singular complex projective algebraic variety $X$ is naturally isomorphic to its $L^2$-
...
5
votes
0
answers
175
views
Is $\overline{\mathcal{M}}_{g,n}$ a Koszul space?
In https://arxiv.org/abs/1902.06318 Dotsenko proved that $\overline{\mathcal{M}}_{0,n+1}$ is a Koszul space, i.e. it is both formal and coformal. Equivalently, a space is Koszul if it is formal and ...
- 641
12
votes
0
answers
482
views
What is the infinity category of subspaces of $\mathbb{R}^n$?
Let $\mathcal{J}$ denote the topological category of finite-dimensional real inner product spaces with linear isometric embeddings. The space of morphisms $\mathcal{J}(\mathbb{R}^k, \mathbb{R}^n)$ is ...
- 901
1
vote
0
answers
111
views
Unique Hausdorff topology on trivial vector bundle?
Question: Is there a Hausdorff topology other than the product topology on $X\times \mathbb{C}^n$, that turns $(X\times \mathbb{C}^n, \mathrm{pr}_1)$ into a vector bundle, where $\mathrm{pr_1}$ ...
- 11
6
votes
0
answers
155
views
How to characterize this condition for commutative squares in $\Delta$
In the simplex category $\Delta$ we have the situation, that
pullbacks exist for cospans $[a] \xrightarrow{\alpha} [n] \xleftarrow{\beta} [b]$ in $\Delta_\text{mono}$ and
pushouts exist spans $[a] \...
- 1,813
62
votes
9
answers
9k
views
There is a nice theory of quadratic forms. How about cubic forms, quartic forms, quintic forms, ...?
Quadratic forms play a huge role in math. This leads one to wonder: Is there a theory of cubic forms, quartic forms, quintic forms and so on? I have failed to discover any. Is there any such theory? ...
- 705
5
votes
1
answer
207
views
Reference for homotopy groups of filtered homotopy colimits
It seems to be well known that for a filtered category $I$
and a functor to the category of pointed spaces $X:I \to \mathcal{S}_*$
the homotopy groups of the filtered homotopy colimit are colimits of ...
- 1,484
13
votes
1
answer
519
views
Low dimensional homotopy groups of $\operatorname{Top}(4)$
$\DeclareMathOperator\Top{Top}$I would like to compute $\pi_3\Top(4)$ and $\pi_4\Top(4)$. It is known that $\Top(4)/O(4) \rightarrow \Top/O$ is 5-connected and
$$
\pi_k(\Top/O) =
\begin{cases}
...
4
votes
1
answer
523
views
Is automorphism on a compact group necessarily homeomorphism? How about N-dimensional torus？ [closed]
Is automorphism on a compact group necessarily homeomorphism? I don't think so，but I think it is possible on the N-dimensional torus.
- 83
8
votes
0
answers
825
views
Can you glue the Betti $X_\text B$ and de Rham stacks $X_\text{dR}$ together?
Let $X$ be a complex algebraic variety. Is there a (derived prestack) over a base
$$\pi\ :\ X_\text{dR,B}\ \to\ S$$
where $S=\mathbf{R},\mathbf{R}/\mathbf{R}^\times,\mathbf{A}^1_\mathbf{C}/\mathbf{G}...
- 5,701
5
votes
2
answers
579
views
Construction of the Mayer-Vietoris spectral sequence
Given subspaces $\{U_i\}_{i \in I}$ of a topological space $X$ with $X = \bigcup_i U_i$ satisfying some conditions, there is a Mayer-Vietoris spectral sequence converging to the homology of $X$. Here ...
- 51
3
votes
2
answers
305
views
Are $H^3(A,U(1))$ and $\operatorname{Ext}^1(A,A^\vee)$ isomorphic for $A$ finite Abelian?
Motivated by three-dimensional Dijkgraaf-Witten TQFTs for finite Abelian groups $A$, that are classified by $H^3(A,\mathbb{R}/\mathbb{Z})$, it seems natural that this group is (naturally) isomorphic ...
- 813
2
votes
0
answers
81
views
A question about the Leray-Serre spectral sequence of the Borel fibration
Let $G$ be a torus which acts on a topological space $X$. Then consider
the Borel fibration $X\longrightarrow X_{G}\longrightarrow B_{G}$. Let $%
\left( E_{r}^{\ast ,\ast },d_{r}\right) $ be the Leray-...
- 1,367
2
votes
0
answers
146
views
Two open contractible subset of $\mathbb{R}^3$ which are not homeomorphic but are polynomial preimage of open sets of $\mathbb{R}$
About 2 decades ago I heared from some one that there are infinitely many open contractible subsets of space mutually non homeomorphic to each other. I confess that I do not remember the ...
- 356
1
vote
2
answers
408
views
Using topology for proving periodicity
Let $f\in\mathcal{C}\big(\mathbb{R},\mathbb{C}^\ast\big)$ a continuous function with modulus $r$ satisfying: $f(t)=r(t)e^{it}$. Assume that the image of $f$ is homeomorphic to the unit circle.
...
- 449
8
votes
1
answer
232
views
Product structure in Milnor exact sequence
Let $h^*$ be a (multiplicative) generalized cohomology theory. Let $X$ be a CW complex which is a union of an increasing sequence $X_0 \subset X_1 \subset X_2 \subset \cdots$ of subcomplexes. Then ...
- 959
7
votes
1
answer
843
views
Algebraic K-theory and Witt groups
Let $S$ be a ring with involution (with 2 invertible). Suppose that the non connective algebraic K-theory $K(S)$ is 0 (i.e. $K_{n}(S)=0$, for all $n$).
Can we say something about the (higher) Witt ...
- 855
18
votes
1
answer
980
views
Are Eilenberg-MacLane spaces limits of manifolds?
In this answer, mme shows that for any compact Lie group $G$, there is a model for its classifying space $BG$ which is the direct limit of closed manifolds. If $G$ is discrete (i.e. $\dim G = 0$), ...
- 19.3k
1
vote
1
answer
385
views
How does homotopy theory simplify topology but allow for complexity in higher category theory?
I'm trying to understand the dual nature of homotopy theory, which seems to play different roles in algebraic topology and higher category theory.
In algebraic topology:
Homotopy theory is often seen ...
- 441
6
votes
1
answer
426
views
Nilpotency of generalized cohomology
$\newcommand\pt{\mathrm{pt}}$Let $(X,\pt)$ be a connected, pointed, finite CW complex and let $h$ be a generalized cohomology theory. Let $\smash{\tilde{h}}^*(X)$ denote the kernel of restriction $h^*(...
- 959
2
votes
0
answers
157
views
Symmetric powers for a short exact sequence of vector bundles
If $0 \to A \to B \to C \to 0$ is a short exact sequence of vector bundles on $X$, is it necessarily true that $[{\rm Sym}^k B] = \sum_{i=0}^k [{\rm Sym}^i A \otimes {\rm Sym}^{k-i} C]$ in the ...
- 2,964
3
votes
0
answers
110
views
String cobracket and co-Hochschild homology
Let $M$ be a closed oriented manifold and take a field of char. zero to be the ground ring. String Topology gives, to the homology $H_\bullet(LM)$ of the free loop space of $M$, the structure of ...
- 985
3
votes
0
answers
89
views
Ordering the elements of a semigroup by $a \le b$ iff $a=b$ or $b=ab=ba$
Let $S$ be a semigroup, written multiplicatively. The binary relation $\le$ on (the underlying set of) $S$, whose graph consists of all pairs $(a,b) \in S \times S$ such that $a = b$ or $b = ab = ba$, ...
- 10.5k
7
votes
4
answers
412
views
Why is the first nontrivial $p$-local stable stem cyclic?
Let $\pi_\ast^{(p)}$ be the ring of $p$-local stable homotopy groups of spheres. This is a nonnegatively graded ring, with $\mathbb Z_{(p)}$ in degree $0$. The first nonvanishing positive degree ...
- 64k
5
votes
1
answer
380
views
Proving the Cork Theorem
I am reading Kirby's paper paper "Akbulut's corks and h-cobordisms of smooth simply connected 4-manifolds" and I have a question about how to actually prove the cork theorem from the results ...
6
votes
0
answers
136
views
Second homotopy group of the symmetric power of a space
Let $X$ be a finite CW complex, $n \ge 2$, and $\Sigma_n$ be the permutation group on $n$ symbols. Let $X^{(n)}=X^n/\Sigma_n$ be the quotient of the natural action of $\Sigma_n$ on $X^n$. We call $X^{(...
- 311
3
votes
1
answer
431
views
Detecting a PL sphere and decompositions
Question 1. A PL $n$-sphere is a pure simplicial complex that is a triangulation of the $n$-sphere. I'm looking for a computer algorithm that gives a Yes/No answer to whether a given simplicial ...
- 903
2
votes
2
answers
185
views
Relative, local coefficient Poincaré duality
Let $X$ be a connected, oriented and aspherical (meaning $\pi_i(X) = 0$ for $i\geq 2$) manifold of dimension $n$, and $S$ a local coefficient system (i.e., a $\pi_1(X)$-module). If $X$ is closed, we ...
- 985
0
votes
0
answers
114
views
Clarifications sought on the paper on the semigroup associated with a free polynomial by Ali Abbas and Abdallah Assi
I have three questions regarding the proof of Proposition 4 on page 4 of this paper here. For those interested in addressing these questions, please refer to some definitions in the first two or three ...
6
votes
1
answer
348
views
Detailed exposition of construction of Steenrod squares from Haynes Miller's book
$\DeclareMathOperator\Sq{Sq}$Chapter 75 of Haynes Miller's book on algebraic topology contains a beautiful construction of the Steenrod squares $\Sq^i$.
Roughly speaking, it goes as follows. All ...
- 63
23
votes
0
answers
717
views
Which proofs of the fundamental theorem of algebra are "essentially the same" vs. "essentially different"?
The classic MO thread Ways to prove the fundamental theorem of algebra contains $60$ proofs of FTA, and I'm sure there are many more in the literature. It would be nice to have some way to organize ...
- 118k
14
votes
1
answer
1k
views
Progress on Gromov's Conjecture of the bound of total Betti numbers
This question is a reference request.
Let $(M,g)$ be a Riemannian manifold of dimension $n$, and $b_i(M) = \dim H_i(M,\mathbb{R})$. Gromov proved it that there are constants $C(n)$ such that, if the ...
- 175
3
votes
0
answers
90
views
Topological groups satisfying the Borel transgression theorem
I am using the Borel transgression theorem as given in Mimura and Toda's "Topology of Lie groups I and II", page 378, Theorem 2.7. I know that it applies when the fiber has the homotopy type ...
- 61
4
votes
3
answers
322
views
Equivariant cohomology of fixed points using the localisation theorem
I am trying to understand the Smith-Thom inequality for spaces equipped with an action by a cyclic group and also the case, when it's an equality:
In the following, let $G=\mathbb{Z}/p$, $\mathbb{F}$ ...
- 131
4
votes
1
answer
514
views
A question about spectral sequences
In the following proof (from The pontrjagin numbers of an orbit map and generalized G-signature theorem by Hsu-Tung Ku & Mei-Chin Ku https://link.springer.com/chapter/10.1007/BFb0085610), it is ...
- 1,367
4
votes
1
answer
218
views
Topological interpretation of the canonical cover of a logarithmic Enriques surface
A normal projective surface $Z$ with at worst quotient singularities is called a logarithmic (log) Enriques surface if its canonical Weil divisor $K_Z$ is numerically equivalent to zero, and $H^1(Z,\...
- 213
6
votes
1
answer
245
views
Fundamental group of the homeomorphism group of a compact manifold
Let $X$ be a compact connected manifold and $\mathcal H(X)$ be the group of all homeomorphisms of $X$, equipped with the compact-open topology. Is the fundamental group of $\mathcal H(X)$ countable? ...
4
votes
0
answers
148
views
Isomorphism between the reduced C*-algebra of a groupoid and the crossed product of inverse semigroups
In Paterson's book "Groupoids, Inverse Semigroups and their Operator Algebras" he proves that for any r-discrete groupoid $G$ with unit space $G^0$, its full $C^* $-algebra $C^* (G)$ is ...
- 261
6
votes
1
answer
206
views
A stable splitting of linear surjections
Some computations I've been doing in Weiss calculus predict the following stable splitting of the space of linear surjections: $\Sigma^\infty_+ \mathrm{Sur}(\mathbb{R}^n,\mathbb{R}^{m_1+m_2})$
as the ...
- 5,849
8
votes
1
answer
485
views
A question about cohomology of the classifying spaces of compact groups
Let $G$ be a compact group (maybe non-Lie group). Let $B_{G}$ denote the
classifying space of $G$. If $G$ contains a circle group $\mathbb{S}^{1}$,
then I think that $H^{\ast }( B_{G};\mathbb{Q}
)$ is ...
- 1,367
1
vote
0
answers
46
views
Optimal transport and the geometry of singular measures on fractal Sets
Let $K$ be a self-similar fractal set in $\mathbb{R}^n$ with Hausdorff dimension $d < n$, equipped with a self-similar measure $\mu$ supported on $K$. Let $\mathcal{P}(K)$ denote the space of ...
- 11
6
votes
0
answers
223
views
Under what generality are the compactly supported singular and sheaf cohomologies equal?
Edit: I have since resolved my question.
If X is locally compact Hausdorff in addition to being cohomologically locally contractible with coefficients in $A$ - eg it is a manifold or an open subset of ...
- 1,020
3
votes
0
answers
134
views
When do quotients of $G$-vector bundles exist?
Let's work in the category of smooth (paracompact, Hausdorff) manifolds. Let $M$ be a manifold and $G$ a Lie group acting on $M$. Suppose $E$ is a $G$-vector bundle on $M$ (that is, $G$ acts on $E$ by ...
- 51
4
votes
0
answers
248
views
Bounds for torsion in Betti cohomology
Let $X\subset \mathbb{P}^{N}_{\mathbb{C}}$ be a smooth, projective variety of dimension $n$ and degree $D$. Is there an upper bound on the torsion in the Betti cohomology groups $H^{i}(X, \mathbb{Z})$ ...
- 41