All Questions
9,056 questions
5
votes
0
answers
160
views
$\infty$-category of spectra and cofibrancy
I have two options for the $\infty$-category of spectra. I would like to know they are equivalent as $\infty$-categories.
Premise: by work of Dwyer and Kan, if we have a simplicial model category, the ...
- 410
3
votes
1
answer
130
views
Why is the Vietoris–Rips complex $\operatorname{VR}(S, \epsilon)$ a subset of the Čech complex $\operatorname{Čech}(S, \epsilon\sqrt{2})$?
$\DeclareMathOperator\Cech{Čech}\DeclareMathOperator\VR{VR}$I am reading Fasy, Lecci, Rinaldo, Wasserman, Balakrishnan, and Singh - Confidence sets for persistence diagrams (see here for a version of ...
5
votes
1
answer
318
views
Surjection onto $H_{2}(\mathrm{PGL}(2,\mathbb{C}),\mathbb{Z})$
Let $G \leq \mathrm{PGL}(2,\mathbb{C})$ be the subgroup of upper-triangular matrices. I am interested in the natural morphism on the Schur multiplier (i.e. group homology as discrete groups)
$H_{2}(G,...
- 305
147
votes
21
answers
23k
views
Are there examples of non-orientable manifolds in nature?
Whilst browsing through Marcel Berger's book "A Panoramic View of Riemannian Geometry" and thinking about the Klein bottle, I came across the sentence:
"The unorientable surfaces are never discussed ...
15
votes
1
answer
787
views
If homotopy groups of spaces are identical, then stable ones are also identical?
Is it true that if pointed spaces $X, Y$ have the same homotopy groups $\pi_n(X) \cong \pi_n(Y)$, then they have the same stable homotopy groups $\pi^S_n(X) \cong \pi ^S_n(Y)$?
In particular, is this ...
- 6,962
4
votes
0
answers
121
views
$E_k$-operads and actions on objects inside $k$-tuply monoidal $n$-category
I understood more or less the claim that $k$-tuply monoidal $n$-categories can be seen as $n$-categories equipped with an action of the $E_k$-operad.
For $k=2$, we have a homotopy equivalence $E_2(r) \...
- 1,813
40
votes
3
answers
7k
views
Timeline of "foundational" advances in homotopy theory?
As an interested outsider, I have been intrigued by the number of times that homotopy theory seems to have revamped its foundations over the past fifty years or so. Sometimes there seems to have been ...
-8
votes
2
answers
861
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 ...
8
votes
0
answers
242
views
Tannaka reconstruction for homotopy types
All sorts of things can be reconstructed from their "linear representations". One example is Tannaka (Deligne, Tannaka-Krein, etc.) reconstruction where a group is recovered from its ...
- 12.3k
0
votes
0
answers
138
views
Shub Conjecture and polynomial entropy
The Shub conjecture on topological entropy $h(f)$ of self map f on manifold M says that the topological entropy is greater (or equal) than (to) the log of maximum absolute values of the ...
- 356
9
votes
1
answer
315
views
Does the Atiyah-Hirzebruch spectral sequence for $E^\ast(X)$ collapse whenever $E$ is complex-oriented and $X$ has even cells?
Let $X$ be a (finite, say — or maybe of finite type) spectrum with even cells (in other words, $H_\ast(X;\mathbb Z)$ is free and concentrated in even degrees). Let $E^\ast$ be a complex-oriented ...
- 64k
0
votes
1
answer
91
views
Topological Properties of Subsets of $R^{m}$ induced by Smooth Manifolds in Matrix Spaces
We know that $M_{m \times n } $ is isomorphic to $R^{mn}$. Let's take a smooth manifold $\mathbb{M}$ in $R^{mn}$ and fix a point in $R^{n}$, say, $p$. Now realize the manifold $\mathbb{M}$ as a subset ...
- 101
24
votes
1
answer
1k
views
What topological principle is at work here?
[I'm cross-posting this from MSE. I initially asked there 10 days ago, and the question was well-received, but left unanswered.]
My question is inspired by a problem I discovered in Putnam and Beyond,...
- 956
7
votes
2
answers
209
views
Cover the $n$-disc irredundantly with $n+1$ open sets. Suppose that the $(n+1)$-fold intersection is empty. Then is some $n$-fold intersection empty?
Cover the $n$-disc with $n+1$ open sets $D^n = U_0 \cup \dotsb \cup U_n$. Suppose that $U_0 \cap \dotsb \cap U_n = \emptyset$. Suppose moreover that the cover is irredundant in the sense that no ...
- 64k
13
votes
1
answer
2k
views
Who proved the motivic 6-functor formalism?
In the recent beautiful talk "Motives and ring stacks" Peter Scholze states the theorem saying that there exists an initial 6-functor formalism on $\mathit{Sch}_\mathbb{Z}$ such that
when $...
- 705
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
6
votes
2
answers
239
views
Euclidean algorithm for simple closed curves
In the proof of Proposition 6.2 in Farb & Margalit, "A primer on mapping class groups", an analog of the Euclidean algorithm is used to construct a simple, closed representative (...
- 83
2
votes
1
answer
423
views
Conjecture about semigroups
Let $G$ be a finite semigroup with order $n$ odd. Let $S_i \in G, i=1,\ldots,\binom{n}{(n+1)/2}$ be all the subsets of $G$ of size $(n+1)/2$.
Let $E(S_i)$ be the set obtained "expanding" $...
- 1,000
4
votes
1
answer
276
views
Preserving simple-connectedness under intersection complexes
Given a simplicial complex $X$, and a family of its subcomplexes $\{U_i\}_{i\in I}$, we define the corresponding intersection complex to be the simplicial complex $X_U$ with vertex set $I$ where $A \...
- 1,926
5
votes
1
answer
430
views
Linking number and intersection number
Consider a disjoint union of two circles $A$ and $B$ smoothly embedded in $\mathbb{R}^3$ with linking number more than $1$. Suppose we know that there exists a disc $D$ in $\mathbb{R}^3$ such that $\...
- 125
6
votes
1
answer
373
views
Fivebrane bordism $\Omega_d^{\mathrm{Fivebrane}}$
$\newcommand{\Fr}{\mathrm{Fr}}\newcommand{\Fivebrane}{\mathrm{Fivebrane}}\newcommand{\String}{\mathrm{String}}\DeclareMathOperator\Spin{Spin}\DeclareMathOperator\SO{SO}\DeclareMathOperator\GL{GL}$What ...
- 10.5k
2
votes
0
answers
84
views
Infinity-morphisms for operadic algebras
Is there an already studied notion of $\infty$-morphism between algebras over a quasi-free operad $P = (T(E), \partial)$?
If the operad $P$ is Koszul, or of the form $\Omega C$ for $C$ a cooperad, ...
- 215
6
votes
0
answers
632
views
Generating functions in countable commutative monoids
Let $f: \mathbb{N}_0 \rightarrow \mathbb{C}$ be a function. The power series of $f$ can be viewed as the function $\mathscr{P}_f : q \mapsto \sum_{n \in \mathbb{N}_0}^{} f(n)q^n$ where $q \in \mathbb{...
- 405
8
votes
1
answer
236
views
Quiver and relations for a monoid related to Catalan numbers
Let $C_n$ be the monoid consisting of monotone maps $\{1,...,n\} \rightarrow \{1,...,n\}$ with $f(i) \leq i$ for all $i$.
The cardinality of $C_n$ is given by the Catalan numbers.
Consider $A_n= \...
- 26.5k
12
votes
1
answer
430
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
8
votes
2
answers
897
views
Can you do geometry with persistent homology?
Setup
In practice, persistent homology of data $X$ is often used to infer the homology of the underlying (Riemannian) manifold $M$ that the data is sampled from.
However most filtrations (Vietoris, ...
- 159
6
votes
1
answer
343
views
Does a ring spectrum with even homotopy and even cells always have a polynomial algebra of homotopy groups?
Let $R$ be a spectrum. Assume that $R$ is bounded-below. Then we can “even-ify” $R$: cone off some generating set of the first nonvanishing odd-dimensional homotopy group of $R$. Do this repeatedly. ...
- 64k
2
votes
1
answer
179
views
Model structures on simplicial presheaves of piecewise-linear manifolds
Let $\mathbf{PL}$ denote the category of piecewise-linear manifolds. The goal is to embed $\mathbf{PL}$ into a category of simplicial presheaves, endow it with a model structure, and then localize it ...
user528837
3
votes
1
answer
246
views
Borel-Moore homology for resolution of singularities
Let $X$ be a singular projective variety. Denote by $Z$ the singular locus of $X$. Consider the resolution of singularities $$\pi: \widetilde{X} \to X$$
Denote by $E$ the exceptional divisor. We know ...
- 2,323
1
vote
1
answer
199
views
Gluing $n$ $2(n-1)$-simplices
It should be true that $n$ $2(n-1)$-simplices can be glued together, such that $k$ of them intersect in a common $(2n -k -1)$-cell and the resulting object is a "convex $2(n-1)$-disc" whose ...
- 1,813
7
votes
0
answers
237
views
Relation beween Chern-Simons and WZW levels, and transgression
3d Chern-Simons gauge theories based on a Lie group $G$ are classified by an element $k_{CS}\in H^4(BG,\mathbb{Z})$, its level. Via the CS/WZW correspondence the theory is related with a 2d non-linear ...
- 813
14
votes
3
answers
652
views
Strøm model structures on the category of simplicial sets
Let $X,Y$ be simplicial sets. A simplicial homotopy is a simplicial map of the form $h:X\times\Delta^1\rightarrow Y$. There are two distinguished maps
$$
in_0:X\cong X\times\Delta^0\xrightarrow{1\...
- 5,596
4
votes
0
answers
107
views
Reference request for a theorem of Jaworowski
Jan Jaworowski, in 2000, proved the following theorem (I came to know about it from here)
Jaworowski (2000) : Let $Y$ be a finite simplicial complex of dimension $k$ and let $n\ge 2k$. If $f:S^n\to Y$...
- 141
1
vote
0
answers
61
views
Map from simplex to itself that preserves sub-simplices: revisited
Here it is proved that, if $f$ is a continuous map from an $n$-simplex $\Delta$ to itself, that maps each sub-simplex of $\Delta$ to itself, then $f$ must be onto $\Delta$ (surjective).
I would like ...
- 3,549
0
votes
1
answer
73
views
If $\widehat{\Gamma}$ is a simply connected clique complex then $\mathrm{Out}(A_\Gamma)$ is an infinite group
Let $\Gamma$ be a simplicial graph and $\widehat{\Gamma}$ be the corresponding clique complex (the flag complex obtained after adding simplices for each compete graph). We can costruct the right-...
- 911
6
votes
1
answer
326
views
Spectral sequence generalizing Čech cohomology
Let $X$ be a 'nice' topological space. Let $\left(U_i\right)_{i\in I}$ be a finite open covering of $X$. Let $\mathcal{F}$ be a sheaf of abelian groups.
For a subset $A\subset I$ denote $$U_A:=\cap_{...
- 21.8k
8
votes
0
answers
230
views
A few questions about Priddy’s construction of $BP$
In A Cellular Construction of BP and Other Irreducible Spectra, Priddy gives an interesting approach to constructing the Brown-Peterson spectrum $BP$. His result is often summarized as
If you start ...
- 64k
2
votes
0
answers
195
views
A $\mathbb{Z}_2$-equivariant map from $n$-torus to $2$-sphere that is null-homotopic is $\mathbb{Z}_2$-homotopic to a non-surjective map?
I have been thinking on the problem below for a while and I am not sure if it is correct or not. I am trying to see if there exists a counter-example for the problem below.
Problem: Let $f: (S^1)^n \...
- 21
7
votes
1
answer
601
views
do all two manifolds admit a three-colorable triangulation?
A triangulation of a two-manifold $M$ is three-colorable if all vertices of the triangulation can be colored red, green, or blue without any two adjacent vertices having the same color.
My question: ...
- 229
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
2
votes
1
answer
57
views
Are simplicial commutative inverse semigroups fibrant?
Let $X$ be a simplicial object in the category of commutative inverse semigroups (or monoids, if needed). Is the underlying simplicial set of $X$ always a Kan complex? If so, are there some nice ...
- 742
3
votes
0
answers
79
views
Rational model for composition of linear isometries
There is a composition map on spaces of linear isometries (over $\mathbb{C}$ say)
$$
\mathcal{L}(\mathbb{C}^k, \mathbb{C}^\ell) \times \mathcal{L}(\mathbb{C}^\ell, \mathbb{C}^m) \longrightarrow \...
- 901
2
votes
0
answers
137
views
details of a dévissage argument for constructible sheaves
I am working on the following Künneth-type isomorphism from [SGA5, exposé III, 2,3]:
$\mathrm{Settings}.$ Let $X_1, X_2$ be separated finite type schemes over the spectrum of a field $S=\mathrm{Spec}...
- 375
8
votes
0
answers
118
views
Defining convex sums locally on the sphere?
$S^1$ and the torus $T^2$ are spaces in which convex combinations don't make sense globally but do locally. Despite their standard representations in $\mathbf{R}^2$ and $\mathbf{R}^3$ respectively not ...
- 637
2
votes
1
answer
95
views
How to determine the LS category of branched covers?
Define the (normalized) Lusternik-Schnirelmann (LS) category of a space $X$, denoted $\mathsf{cat}(X)$ to be the least integer $n$ such that $X$ can be covered by $n+1$ number of open sets $U_i$ each ...
- 31
3
votes
0
answers
93
views
References for variations of Seifert–van Kampen's theorem: HNN extensions and "sensible" intersections
A basic consequence of the Seifert–van Kampen theorem is the following.
Theorem: Consider a union of topological spaces $X$, $Y$ whose intersection $X\cap Y = Z$ is open connected and $\pi_1$-...
- 1,033
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
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
11
votes
1
answer
658
views
Can you deduce the correspondence between 2D oriented TQFTs and commutative Frobenius algebras from the (framed) Cobordism Hypothesis?
Background
I am currently writing an MSc dissertation on TQFTs (and Khovanov homology, but that is unrelated to this question).
After having read most of Kock's book on the equivalence between 2D ...
3
votes
1
answer
238
views
Steenrod operations on classifying spaces
Steenrod operations can be defined for all finite characteristics $p$. The simplest one, when $p=2$, is the Steenrod square. I wonder if the computation for classifying spaces for classical Lie groups ...
- 803