All Questions
8,725 questions
2
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109
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Whitehead lemma for simplicial Lie algebras
Let $R \to R'$ be a morphism of connected free simplicial Lie algebras. Then the analog of the Whitehead lemma states that if $\pi_* f_{\mathrm{ab}}$ is an isomorphism then $\pi_* f$ is an isomorphism....
- 41
8
votes
2
answers
600
views
Derivations in the Steenrod algebra
Let $\mathcal A^\ast$ be the (mod 2) Steenrod algebra.
Question 1:
Is there a classification of homogeneous elements $D \in \mathcal A^n$ such that $D^2 = 0$?
Question 2: Is there a classification of ...
- 64k
9
votes
1
answer
326
views
What is the center of Morava $K$-theory?
Let $E$ be an $E_1$ ring spectrum. Then I believe the center of $E$ is an $E_2$ ring spectrum over which $E$ is an $E_1$ algebra, given by the endomorphisms of $E$ as a bimodule over itself.
Question: ...
- 64k
4
votes
0
answers
184
views
Obstruction to finding a Whitney disk
Let $p,q\geq 3$ be integers. Let $M$ be a compact oriented smooth $(p+q)$-manifold and $P$ and $Q$ compact submanifolds of dimensions $p$ and $q$ intersecting transversely. Assume that $M,P$ and $Q$ ...
2
votes
1
answer
482
views
homotopic to a constant map
Let $X$ and $Y$ be topological spaces and more precisely connected finite CW complexes.
Let $f\colon X \to Y$ be a continuous map such that there exist a second continuous map $F\colon X^3 \to Y$ and
$...
- 185
1
vote
0
answers
153
views
Stable homology of general linear groups
For what class of rings $R$, is the stable homology (with various choices of coefficients) of $GL_n(R)$ known? Borel computed it rationally for number rings, Quillen computed it for finite fields. Are ...
- 965
8
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0
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302
views
Can Postnikov towers converge without Postnikov completeness?
In Higher Topos Theory, Lurie says that "Postnikov towers are convergent" in a presentable $\infty$-category $\mathcal{C}$ if $\mathcal{C}$ is equivalent to the $\infty$-category $\mathrm{...
- 25.2k
12
votes
1
answer
379
views
Approximate classifying space by boundaryless manifolds?
As pointed out by Achim Krause, any finite CW complex is homotopy equivalent to a manifold with boundary (by embedding into $\mathbb R^n$
and thickening), and so every finite type CW complex can be ...
- 123
5
votes
0
answers
120
views
Torus equivariant Morava K-theory
Let $X$ be a CW complex with a torus action $T$. Is there an established definition in equivariant stable homotopy theory of $T$-equivariant Morava K-theory, $K_p(n)^*_T(X)$? Any explicit references ...
- 959
5
votes
0
answers
110
views
Are there exotic examples of a Lie group up to coherent isotopy?
This question is based on attempting to construct the (homotopy type) of Lie groups using Cobordism Hypothesis style abstract nonsense.
There is an $\infty$-groupoid of smooth, framed manifolds where ...
- 563
13
votes
1
answer
580
views
Identifying two definitions of orientation on a vector space
Let $V$ be an $n$-dimensional real vector space. Here are two definitions of an orientation on $V$:
A generator of the $1$-dimensional vector space $\wedge^n V$, up to multiplication by positive ...
- 133
3
votes
1
answer
149
views
(Derived category of) sheaves over an infinite union
The short version of my question is:
Suppose $X$ is a (reasonably nice) topological space such that $X = \bigcup_{n \ge 1} X_n$ for an increasing sequence of (closed) subspaces $X_1 \subset X_2 \...
- 197
1
vote
0
answers
201
views
An open ended question: The dual of a covering map? Is this a real thing?
Reposted from this Reddit post as I didn't get good answers there:
So I've been reading about the Galois theory of covering maps and been staring at this equation for way too long:
$$\left| \pi_1(X,...
- 19
2
votes
0
answers
122
views
Quasi-isomorphisms of P-algebras
In the paper "Homotopy algebras are homotopy algebras" from Markl a notion of strong homotopy morphism between strong homotopy P-algebras is defined. The author restricts to the case where $...
- 215
4
votes
0
answers
79
views
On the induction step in Theorem 2.6 of "Homological stability for linear groups" by Kallen
I am currently reading the proof of the connectivity theorem of Wilberd van der Kallen (Theorem 2.6 in https://link.springer.com/article/10.1007/BF01390018) for a seminar talk. I am a little stuck on ...
- 71
4
votes
2
answers
287
views
Teaching suggestions for Kleene fixed point theorem
I will take over two lectures from a colleague in which we discuss fixed point theory in the context of complete partial orders, and culminates in showing the Kleene fixed point theorem (see f.e. ...
- 41
4
votes
1
answer
274
views
Comparing Kummer maps to étale homotopy at finite level
$\DeclareMathOperator\Mor{Mor}\DeclareMathOperator\Hom{Hom}\newcommand{\et}{\mathrm{et}}\newcommand{\top}{\mathrm{top}}$In Voevodsky's paper "Étale topologies of schemes over fields of finite ...
- 544
3
votes
1
answer
124
views
Extending curves on a surface to a basis for its first homology satisfying intersection criteria
The title suggests a broader scope of inquiry, but my question mostly pertains to the following example:
Let $(Y, \mathcal{Z}, \phi)$ be a bordered 3-manifold with Heegaard diagram $\mathcal{H}$ of ...
- 131
-8
votes
2
answers
862
views
Homotopy theory and algebraic topology last 10 years. Is it a dying field? [closed]
I'm under the impression that algebraic topology is a dying field in mathematics. That was my impression but I think I'm wrong. As every person I do need some evidence that my impression is not ...
0
votes
1
answer
155
views
Vector bundles over a homotopy-equivalent fibration
I think this question is related to what is known as "obstruction theory", but I'm not very familiar with this field of mathematics, so I am asking here.
Let $\pi:N\rightarrow M$ be a smooth ...
- 3,307
5
votes
3
answers
497
views
Bar construction in commutative algebras is calculated by pushout
$\DeclareMathOperator\colim{colim}$
Also asked in MathStackexchange here
This is a statement in Lurie's Higher Algebra 5.2.2.4.
Proposition 3.2.4.7 in HA said that the monoidal structure on $\text{...
- 163
9
votes
1
answer
325
views
$\operatorname{Spaces}/BG$ $\sim$ $\operatorname{Spaces}^G$ $\sim$ $??(\Omega G)$
This is a crosspost (with minor alterations).
For a topological group $G$, assigning to a $G$-space $X$ the (canonical) map $EG\times_GX\to BG$ establishes an equivalence between the homotopy category ...
- 18.4k
0
votes
1
answer
94
views
Homology of independence complex after removing a vertex
Let $G$ be a chordal graph, $I(G)$ be its independence complex, and $v \in V(G)$ be a simplicial vertex (that is, $v$'s neighborhood is a clique).
Is there a way to relate the homology of $I(G)$ and ...
- 105
2
votes
1
answer
199
views
Regular sequence in cohomology of Grassmannians
$\DeclareMathOperator\Gr{Gr}$Consider the polynomial ring $\mathbb{Z}[x_1,\dots,x_m, y_1,\dots,y_n]$, I want to prove that the sequence $$x_1 + y_1, x_2 + x_1y_1 + y_2, \dots, x_my_{n-1} + x_{m-1}y_n, ...
- 185
7
votes
1
answer
101
views
Bisimplicial spaces as a coequalizer of maps between "simpler" bisimplicial spaces
From a bisimplicial space $T$, one can consider the simplicial spaces $p \mapsto T_{pp}$, $p \mapsto | q \mapsto T_{pq}|$, and $q \mapsto |p \mapsto T_{pq}|$, where $| \cdot|$ denotes geometric ...
- 71
0
votes
0
answers
86
views
Projectivity of equivariant K-theory of toric variety
I'm looking at Vezzosi and Vistoli's paper: Higher algebraic K-theory for actions of diagonalizable groups.
In Theorem 6.9, they prove that the $T$-equivariant K-theory of a smooth projective toric ...
- 959
8
votes
0
answers
155
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Cohomology dimensions are well approximated by Gaussian for multiply-fibered manifolds ? (Topological central limit theorem)
Consider some manifold $M$ say compact smooth. Let $b_i$ be its Betti numbers (non-zero), i.e. its cohomology dimensions.
Assume $M$ can be subsequently fibered by many manifolds, i.e. there is $ M_{...
- 24.9k
0
votes
1
answer
170
views
Isn't every algebraic operad equipped with a trivial weight?
In Loday–Vallette "Algebraic Operads" they state the following result (Theorem 6.6.2, Operadic twisting morphism fundamental theorem):
Let $P$ be a connected weight graded differential ...
- 215
4
votes
1
answer
215
views
Co-index of a Space
I am going through this paper by Tanaka, but I got stuck at Proposition 2.4, given below.
He does not provide any proof, instead referring to Theorem 6.6 of this paper, given below.
Unfortunately, I ...
0
votes
0
answers
150
views
Connectedness of deleted symmetric product
Let $X$ be a connected Hausdorff space. It is well-known that the $n$-fold symmetric product $\mathcal{F}_n(X) := \{A\subseteq X : 0<|A|\leq n\}$ is a connected space equipped with the Vietoris ...
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2
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0
answers
414
views
$$ \left(\frac{\text{Man}^{\text{fr}}}{\text{Cobordism}},\coprod,\times \right)\simeq \left((\text{Fin}^{\simeq},\coprod)^{\text{gp}},\times\right)?$$ [closed]
If we combine a theorem of Pontryagin and the Barratt-Priddy-Quillen theorem we get that both sides of
$$
\left(\frac{\mathrm{Man}^{\mathrm{fr}}}{\mathrm{Cobordism}},\coprod,\times \right)\simeq \left(...
- 705
15
votes
0
answers
317
views
Does there exist a closed 4-manifold whose $\pi_2$ contains torsion elements
Does there exist a closed 4-manifold $X$ such that $\pi_2(X)$ contains torsion elements?
And, if so, does there exist a closed 4-manifold $X$ such that $\pi_2(X)\neq 0$ but $\pi_2(X)\otimes \mathbb{Q}=...
- 151
5
votes
1
answer
436
views
Does coproduct preserve cohomology in differential graded algebra category
Consider two cochain DGA (differential graded algebras) named $A$ and $B$. By "coproduct" of two DGA I mean the category theory coproduct, not the coalgebra coproduct. It is defined in "...
- 159
2
votes
1
answer
179
views
Factorization systems for vector bundles
Are there any well-known factorization systems for the category of vector bundles defined over topological spaces?
- 615
5
votes
0
answers
192
views
When is the classifying space of a group/H-space rationally equivalent to a product of Eilenberg-MacLane spaces?
The MO-question asks why the classifying space of a group is not necessarily rationally a product of Eilenberg–MacLane spaces.
I am looking for classes of examples of connected topological groups/...
- 1,174
0
votes
0
answers
101
views
Finding an example if it exists, for a non-contractible and contractible space with special requirement on quotients of their union?
Let $A$ and $B$ be subsets of $n$-dimensional Euclidean space $\mathbb{R}^{n}$, such that $A$ is non-contractible, $B$ is contractible and $B$ is not an one-point set.
I would like to find example(s) ...
3
votes
1
answer
263
views
Original proof of Lefschetz's theorem on $(1,1)$ classes
Is there a "modern" account of Lefschetz proof of his theorem about $(1,1)$ classes for projective surfaces ?
I believe that would be very interesting to understand the original arguments ...
12
votes
1
answer
433
views
Plus construction on Simplicial Sets?
I had asked this question in Math StackExchange a few days ago, but didn't get any answers. I believe its more suitable to be asked here.
Write $\mathsf{sSet}$ for the category of simplicial sets and $...
- 174
2
votes
0
answers
112
views
A cell complex constructed from singular knots
Let $\mathcal K_n$ be the set of all $n$-singular knots up to isotopy,i.e. an immersion of $S^1$ into $\mathbb R^3$ with $n$ transverse double points that is an embedding when restricted to the ...
- 141
5
votes
2
answers
583
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
13
votes
1
answer
2k
views
Roadmap for Algebraic Geometry/Homotopy Theory/Algebraic $K$-Theory intersection
I’m afraid that this is quite a general question but I am hoping some experts can weigh in. I am a student generally interested in learning more about the intersection of algebraic geometry, algebraic ...
- 155
4
votes
2
answers
344
views
Lifting of map from $S^3$ to itself
My question concerns the lifting of degree $0$ map from $S^3$ to itself.
Let us suppose that all maps are smooth here.
Looking at $S^3$ as the space of unit quaternions, one way to define degree is ...
- 914
6
votes
0
answers
122
views
Explicit representatives for Borel cohomology classes of a compact Lie group?
I'm looking for explicit representatives of $H^3_{Borel}(G, R/Z)$, i.e. a measurable function $G^3\to R/Z$ representing a generator of the cohomology group. (Here $G$ is a compact (perhaps simple) ...
- 12.8k
2
votes
1
answer
215
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Is the category of simplicial $R$-modules closed monoidal?
I am trying to understand if the simplicial mapping space for simplicial $R$-Modules (or at least simplicial vector spaces) is adjoint to the (level-wise) tensor product. For simplicity, let me state ...
- 91
1
vote
1
answer
144
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About Čech cohomology in transformation groups
I'm starting a study about theory of transformation groups and equivariant cohomology, in what I read several times that Čech cohomology is the most compatible with this theory, but until now I haven'...
- 237
4
votes
0
answers
453
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Problem 1.8 from Kirby's list
Context
I looked through a book called "Problems in Low-Dimensional Topology", where Rob Kirby lists a set of problems. He provides a list of problems, states their conjectures, and ...
0
votes
0
answers
147
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Noether's formula for real algebraic surfaces
Is there a version of Noether's formula for the Euler characteristic of a surface for Real algebraic surfaces?
Specifically, given $X$ a real algebraic compact smooth surface, what is the relationship ...
2
votes
1
answer
628
views
Does some published texbook take a particular approach (described here) to the transition from discrete to continuous probability distributions?
(I posted this question at matheducators.stackexchange.com and it seems to be considered an inappropriate question for that site. I don't understand why.)
Imagine an introductory probability course ...
7
votes
1
answer
191
views
Reference request for equivalences between different models of lax limits
There are several models for lax limits of model categories/ $\infty$-categories in the literature. For example, within the realm of $\infty$-categories one can construct them using coCartesian ...
- 177
2
votes
0
answers
117
views
Hypercube of chain complexes as functor from (Δ^1 )^n to ∞-category of chain complexes
A hypercube of chain complexes consists of $\mathbb{Z}$-graded vector spaces $C_\epsilon$ for $\epsilon\in\{0,1\}^n$ and maps $D_{\epsilon,\epsilon^\prime}:C_{\epsilon}\to C_{\epsilon^\prime}$ for $\...
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