All Questions
2,364 questions with no upvoted or accepted answers
10
votes
0
answers
546
views
When is the one-point compactification well-pointed?
This is a follow up to my previous
question.
Question:
Is there a reasonably natural set of conditions which guarantee that the one-point
compactification $X^+$ of a locally compact Hausdorff ...
- 18.9k
10
votes
0
answers
1k
views
Two model categories I would like to know if they are Quillen equivalent or not
It is the motivation of the question Examples of non Quillen-equivalent model categories having equivalent homotopy categories. I did not give at first the motivation because i don't think that people ...
- 3,901
10
votes
0
answers
740
views
What is Quillen's contribution to index theorem?
In the book "Heat Kernels and Dirac Operators" by Berline, Getzler and Vergne it is said that "Our book is based on a simple principle, which we learned from D. Quillen: Dirac operators are a ...
- 9,019
10
votes
0
answers
281
views
Embeddings of hyperbolic $n$-manifolds in $R^{n+2}$
Is there any example of a compact manifold $M$ of dimension $n>10000$
such that
$M$ admits an embedding into $\mathbb R^{n+2}$,
$M$ is hyperbolic; i.e., it admits a Riemannian metric with
...
- 28.9k
10
votes
0
answers
493
views
Differentials in the Adams-Novikov spectral sequence and the geometric boundary theorem
$\newcommand\Ext{\mathrm{Ext}}
\newcommand\Z{\mathbb{Z}}
\newcommand\G{\mathbb{G}}$
The reference for this question will be the paper by Henn, Karamanov and Mahowald - "The homotopy of the $K(2)$-...
- 3,784
10
votes
0
answers
235
views
Computation of stable cohomology ring of SL_n(Z) using algebraic topology
It is known that $H^k(SL(n,\mathbb{Z}))$ is independent of $n$ for $n \gg k$, so we can define a stable cohomology ring
$$V = \text{lim}_{n \rightarrow \infty} H^{\ast}(SL(n,\mathbb{Z});\mathbb{R}).$$...
- 101
10
votes
0
answers
227
views
Spaces with free MU-homology
Let $E$ be a homology theory whose coefficients $E_*(pt)$ are concentrated in even dimensions. This could be complex bordism $MU$, but also complex $K$-theory, $BP\langle n \rangle$, $E(n)$, ...
Let ...
- 12.1k
10
votes
0
answers
458
views
is a group $G$, that admits finite $k(G, 1)$ and has no Baumslag-Solitar subgroups, necessarily hyperbolic?
This is the first question asked in Bestvina's article "Questions in Geometric Group Theory". Does anyone know if there has been any progress made on this problem? Is the question answered if $G$ is ...
- 101
10
votes
0
answers
325
views
H-space structure on the Calkin algebra
By the Atiyah-Jänich theorem the K-group $K^0(X)$ for a compact space $X$ may be represented as $[X, U(Q)]$, where $Q = B(H)/K(H)$ is the Calkin algebra and $H$ is a separable infinite dimensional ...
- 7,637
10
votes
0
answers
464
views
Bicommutative Hopf algebras have internal hom objects. What are they?
Let $k$ be a field and $\mathcal H$ the category of conilpotent bicommutative Hopf algebras over $k$. (Conilpotent means that every element eventually becomes zero if you apply the reduced ...
- 6,162
10
votes
0
answers
362
views
Question about $A_\infty$ maps
Given $A_\infty$-spaces $X$ and $Y$, Boardman and Vogt defined an $A_\infty$-map from $X$ to $Y$ to be a map $f: X \to Y$ of underlying based spaces and an $A_\infty$-structure on the reduced mapping ...
- 18.9k
10
votes
0
answers
484
views
Applications of sheaf theory to the computation of invariants of LS-category type
I would like to know if sheaf theory can be applied to a particular class of questions in topology.
The Schwarz genus (also known as sectional category) of a continuous map $p\colon\thinspace E\to ...
- 35.9k
10
votes
0
answers
735
views
Adams Spectral Sequence for Equivariant Cohomology Theories
In ordinary algebraic topology the Adams spectral sequence can be applied for any cohomology theory $E$ and in good cases it converges to the stable homotopy classes of maps (of the E-nilpotent ...
- 1,273
9
votes
0
answers
160
views
Is there a closed aspherical manifold with infinitely many symmetries and without essential immersed tori?
The precise question is the following:
Is there a closed aspherical manifold $M$ of dimension $n\geq 3$ such that Out($\pi_1(M)$) is infinite and $\pi_1(M)$ does not contain $\mathbb Z \times \mathbb ...
- 10.5k
9
votes
0
answers
120
views
Reference Request: Moore--Postnikov tower of the rationalization of a fibration
Two spaces $X$ and $Y$ are said to be rationally homotopy equivalent, written $X \sim_{\mathbb{Q}} Y$, if their rationalizations $X_{\mathbb{Q}}$ and $Y_{\mathbb{Q}}$
are homotopy equivalent. Moreover,...
- 371
9
votes
0
answers
212
views
Left adjoint functor between categories of polygons?
EDIT: Based on very helpful comments from Alec Rhea and Qiaochu Yuan I am adding some specification on objects and morphisms, hoping that this clarifies the idea behind these categories. I have also ...
- 6,937
9
votes
0
answers
147
views
degree 1 maps for bordism homology
Let $f\colon X \to Y$ be a degree 1 map between closed oriented manifolds. Then the induced homomorphism between the homology groups is surjective up to torsion.
Can one say something similar about (...
9
votes
0
answers
569
views
The relation between the motivic Galois group and the motivic Steenrod algebra
There is a point of view on the Steenrod algebra that goes something like the following: the functor $-\otimes H\mathbb{F}_p\colon Mod_{\mathbb{S}}\to Mod_{H\mathbb{F}_p}$ corresponds to pulling back ...
- 10.5k
9
votes
0
answers
182
views
What homology theory is calculated by unreduced cubical chains?
For a topological space $X$ and a subspace $A$, let $Q_n(X,A)$ be the group of singular cubical $n$-chains of $X$ relative to $A$ and let $D_n(X,A)$ be the subgroup of degenerate cubical chains. The ...
- 91
9
votes
0
answers
269
views
Colimits of symmetric groups
The infinite symmetric group $S_{\infty}$ of finitely supported permutations of $\mathbb{N}$ can be written as a colimit over the $S_n$'s with respect to the embedding $S_{n} \to S_{n+1}$ that maps $\...
- 7,637
9
votes
0
answers
195
views
Every locally presentable $\infty$-category can be presented by a proper model category
Is there an argument in the litterature that show that every locally presentable $\infty$-category is equivalent to the localization of proper combinatorial Quillen model category ?
Of course if one ...
- 42.4k
9
votes
0
answers
405
views
What is the Balmer spectrum of the p-complete stable homotopy category?
When doing computations with spectra, we first reduce to working at a prime p by using the arithmetic fracture theorem: (the homotopy groups of) a spectrum of finite type can be recovered from its ...
- 1,969
9
votes
0
answers
322
views
Chromatic filtration on stable homotopy
Chapter 5 of Ravenel's green book starts with the sentence “[The chromatic spectral sequence] is a mechanism for organizing the Adams-Novikov $E_2$ term and ultimately $\pi_*(S^0)$ itself." My ...
- 133
9
votes
0
answers
212
views
Is the category of all topological spaces, including the bad ones, simplicially tensored and cotensored?
Let $\textbf{Top}$ be the category of all topological spaces, including the bad ones.
We can make $\textbf{Top}$ into a simplicially enriched category as follows:
Given topological spaces $X$ and $Y$,...
- 15.9k
9
votes
0
answers
189
views
Operads and Inverses
One slightly displeasing thing about the theory of operads is that they are unable to encode the structure of inverses; e.g. the recognition principle for $n$-fold loop spaces says "there is an ...
- 423
9
votes
0
answers
341
views
Is an $n$-connected space a homotopy colimit of $n$-connected spheres?
A pointed, $n$-connected space $X$ admits a CW structure using just $n$-connected spheres, so $X$ is an iterated homotopy colimit of $n$-connected spheres (i.e. you start with spheres and keep taking ...
- 64k
9
votes
0
answers
205
views
Donaldson invariants for piecewise-linear $4$-manifolds
It is well known that in dimension $4$, the notion of piecewise linear manifolds and the notion of smooth manifolds are the same [1][2]. On the other hand, the computations of Donaldson invariants ...
- 5,230
9
votes
0
answers
519
views
Why is the symbol map in Atiyah–Singer paper continuous?
I am reading "Index of elliptic operators: I", by Atiyah and Singer these days and I am trying to understand all the paper. I find it difficult to verify the following statement on page 512:...
9
votes
0
answers
186
views
Aspherical fibrations and group epimorphisms
Let $\mathsf{Top}$ denote the category of pointed spaces having the pointed homotopy type of a pointed CW-complex. Let $\mathsf{Grp}$ denote the category of groups. It is well documented that for ...
- 35.9k
9
votes
0
answers
308
views
Refinement of hypercovers by ordinary covers
I am asking for references and discussions of statements of the form
Every bounded hypercover can be refined by an ordinary cover
By "bounded" I mean "finite height". E.g., are ...
- 4,519
9
votes
0
answers
228
views
What is the operad for homotopy associative, homotopy commutative objects?
There is an operad whose algebras are objects with a homotopy unital multiplication -- the $A_2$ operad.
There is an operad whose algebras are objects with a homotopy unital, homotopy associative ...
- 64k
9
votes
0
answers
235
views
A conjecture of Adams on the mod p cohomology of classifying spaces of Lie groups
In the paper On the mod p cohomology of BPU(p), the authors say that there is a conjecture of J. F. Adams as follows:
Conjecture (J. F. Adams) Let $G$ be a compact connected Lie group, and let $p$ be ...
- 935
9
votes
0
answers
124
views
An overcomplex of the Cartan model in equivariant cohomology?
Given any graded vector space $V^\bullet$ and any degree 1 linear operator $d\colon V^\bullet\to V^{\bullet+1}$, one gets a complex $(\ker(d^2),d)$. Moreover, if $V^\bullet$ is a graded algebra and $d$...
- 6,777
9
votes
0
answers
196
views
Leech lattice and rational varieties
Question: Is there a smooth rational variety $X$ of complex dimension $4n$, $n \in \mathbb{N}$; such that the intersection form on $H^{4n}(X,\mathbb{Z})$ is the Leech lattice?
My motivation is mainly ...
- 6,995
9
votes
0
answers
372
views
Groups with trivial outer automorphism group and prescribed center?
Given an arbitrary abelian group $A$, can we find a group $G$ such that
$\mathrm{Out}(G)=\mathrm{Aut}(G)/\mathrm{Inn}(G)=1$, and
$Z(G)\simeq A$?
Why is this interesting? Given a group $G$, we have ...
- 27.2k
9
votes
0
answers
333
views
Homotopical characterization of CW complexes
Let $X$ be a compact metrizable topological space of covering dimension $n\leq 3$.
Is it possible to give a necessary and sufficient condition for $X$ to be a CW complex in terms of the homotopy types ...
- 117
9
votes
0
answers
200
views
Homotopical characterization of manifolds
Let $X$ be a compact metrizable topological space of covering dimension $4$.
Assume that for any point $x\in X$ any neighbourhood of $x$ contains a contractible open neighbourhood $U$ such that $U\...
- 117
9
votes
0
answers
361
views
Bernoulli-like polynomials
Let $\psi_0 (x,t)=\frac{te^{xt}}{1-e^{-t}}$. Then
$$\psi_0(0,t)=\frac{t}{1-e^{-t}};$$
$$\psi_0(x,t)=1+\sum_{n=1}^\infty \frac{t^n}{n!} B_n(x)$$
where $B_n$ is a monic polynomial of degree $n.$
Now ...
- 431
9
votes
0
answers
393
views
When is an increasing union a colimit?
Let's consider a diagram $\Phi: \lambda \to \mathcal{T}_*$
$$
X_0 \to X_1 \to \cdots \to X_\xi \to X_{\xi+1} \to \cdots
$$
of pointed spaces,
indexed by some ordinal $\lambda$, in which each $X_\xi$ ...
- 12.5k
9
votes
0
answers
338
views
Is $\beta^{*}(w_{2k-2}) = 0$ for an open orientable $2k$-manifold?
This question is motivated by the vector field question I asked recently. Panagiotis Konstantis answered this question for odd manifolds and I am trying to figure out the even case.
Let $M$ be a ...
- 1,726
9
votes
0
answers
216
views
Some computational results and goals of stable motivic homotopy theory of schemes
I am trying to learn ($\mathbb{P}^1$-)stable motivic ($\mathbb{A}^1$-)homotopy theory of schemes from the Cisinski-Deglise book, Triangulated Categories of Mixed Motives. In order to keep myself going ...
- 91
9
votes
0
answers
287
views
Rational cobordism classes of manifolds fibered over spheres
Let us fix positive integers $k, m$. Let $A^k_{4m} \subset \Omega^{\text{SO}}_{4m} \otimes \mathbb Q$ be the subgroup generated by oriented cobordism classes of manifolds fibered over $S^k$.
The ...
- 11.9k
9
votes
0
answers
331
views
What is the current state of research in Chern-Simons theory?
I'm a PhD student in mathematical physics looking for some orientation. As asked in the title, I would like to know the current state of research in Chern-Simons theory. More specifically, what are ...
- 131
9
votes
0
answers
887
views
How many ways are there to teach class field theory?
I will soon have to teach class field theory (I do not know whether it will be local or global yet:)) to postgraduate students. I wonder, which approaches to this subject(s) exist now.
I definitely ...
- 16.9k
9
votes
0
answers
317
views
Are there non-obvious finite $E_\infty$ ring spectra?
I see two "obvious" classes of nonzero finite $E_\infty$ ring spectra $R$:
$R = \Sigma^\infty_+ (S^1)^{\times n}$
$R = D\Sigma^\infty_+ X$ ($X$ a finite space)
Questions:
Are there any others?
In ...
- 64k
9
votes
0
answers
698
views
Power maps of contractible topological groups
I'm little bit puzzled with the following question, I was wondering if someone could help with a suggestion or a hint.
Let start with some notations. Suppose that $G$ is a Hausdorff topological ...
- 1,974
9
votes
0
answers
344
views
Example of a finitely-dominated CW complex which is not finite?
It's easy to find sources discussing Wall's finiteness obstruction, but harder to find discussion of examples. What's an example of a finitely-dominated CW complex which is not homotopy equivalent to ...
- 64k
9
votes
0
answers
401
views
Higher homotopy groups of Calabi-Yaus
Is something known about the higher homotopy groups of Calabi-Yau threefolds? For example, one of the easiest CYs is the quintic, defined as the anticanonical divisor in $\mathbb{CP}_4$. What are its ...
- 489
9
votes
0
answers
499
views
About Kan-Thurston theorem
The Kan-Thurston theorem obsesses me recently, which is proved by Daniel Kan and William Thurston, Every connected space has the homology of a K(π,1), Topology Vol. 15. pp. 253–258, 1976. Later, there ...
- 1,906
9
votes
0
answers
207
views
An equivalent definition for $\text{Spin}^c$-structures
I'm interested in proving the following proposition ([G], Remark page 48):
Prop: A $\text{Spin}^c$-structure over an oriented vector bundle is equivalent (after stabilizing if the fiber dimension ...
- 2,018