All Questions
2,364 questions with no upvoted or accepted answers
12
votes
0
answers
104
views
How many upper sets in this decomposition of finite posets
Let $X$ be a finite poset.
If
$$X = X_1 \cup X_2$$
where $X_1$ and $X_2$ are strict upper sets, then a lot of properties of $X$ can be inferred from the smaller posets $X_1, X_2$ and $X_1\cap X_2$ (...
- 221
12
votes
0
answers
659
views
Understanding a certain algebraic set arising in Deep Learning
I'm not a professional geometer. Thanks in advance for your patience.
So, let $n$, $k$, $p_0,\ldots,p_{k}$ be positive integers. Let $X$ (resp. $Y$) be an $p_0$-by-$n$ (resp. an $p_{k}$-by-$n$) real ...
- 6,853
12
votes
0
answers
919
views
Are the Alexander-Whitney and Eilenberg-Zilber maps homotopy inverse in arbitrary abelian categories?
Let $\mathcal{A}$ be a monoidal abelian category. Let $A$ and $B$ be simplicial objects in $\mathcal{A}$, and let $N_\ast(-)$ denote the normalized chain complex functor. Let
$$AW_{A,B}\colon N_\ast(...
- 721
12
votes
0
answers
519
views
Does Lefschetz-type theorems imply ampleness?
Let $X$ be a smooth $n$-dimensional complex projective variety and $D \subset X$ a smooth (effective) divisor. Consider the following properties:
$D$ is ample.
(Positivity) For any $k$-dimensional ...
- 1,170
12
votes
0
answers
905
views
Conditions for the second homotopy group to be Abelian
What is the weakest set of assumptions on a pair of spaces $X\subset M$ for which the second homotopy group $\pi_2(M,X) $ is guaranteed to be Abelian?
Naively, I expected that Abelian $\pi_1(X)$ ...
- 403
12
votes
0
answers
267
views
Simply connected homology cobordisms
I'm looking for interesting examples of a homology 3-sphere $Y$ for which there exists a smooth, simply connected homology cobordism from $Y$ to itself (or simply to another homology 3-sphere $Y'$, ...
- 779
12
votes
0
answers
306
views
Understanding a formula in Ozsvath-Szabo
I'm a beginning graduate student reading Ozsvath-Szabo's foundational paper, Holomorphic disks and topological invariants for closed 3-manifolds. What I have trouble understanding is a formula in ...
- 355
12
votes
0
answers
278
views
Is there a geometric interpretation of the cohomology of an automorphism group acting on a universal deformation ring?
Let $X_0$ be some algebro-geometric object defined over a field, and suppose its deformation functor is prorepresentable, so there is a universal deformation ring $R$. Then $Aut(X_0)$ acts naturally ...
- 10.7k
12
votes
0
answers
313
views
For a Banach space $X$, when is $X$ homeomorphic to $X \setminus A$?
$\mathbb{R}^n\not\cong\mathbb{R}^n\setminus\{0\}$ are not homeomorphic is a triviality from Algebraic Topology. On the other hand, if $X$ is an infinite dimensional Banach space, then $X \cong X\...
- 43.2k
12
votes
0
answers
551
views
Goodwillie's notes from MSRI Lecture Series
Does anyone know where I can find an electronic version of Goodwillie's (unpublished) notes from the MSRI Lecture Series in Spring, 1990? They're mentioned/cited as such in work of Dundas-McCarthy, ...
12
votes
0
answers
695
views
"To operate the machine, it is not necessary to raise the bonnet."
The quotation in the title is attributed to Frank Adams and appears in several places:
In the preface of [2002, Operads in algebra, topology and physics]: "to operate the machine, it is not necessary ...
- 15.9k
12
votes
0
answers
716
views
The general Smith homomorphism in bordism
The Smith homomorphisms are a family of homomorphisms between equivariant bordism groups in different dimensions.
One example that is known to be an isomorphism is the map
$$
\tilde\Omega_{d+1}^{\rm ...
- 631
12
votes
0
answers
656
views
Has this chain complex associated with a simplicial complex been studied before?
I have stumbled upon a construction that has probably been noticed before, and I wonder if anyone can point me to a reference.
Suppose that $K$ is a simplicial complex. Let $P(K)$ be the free abelian ...
- 55.9k
12
votes
0
answers
747
views
Cohomology and impossible figures
In connection with the MO question Occurrences of (co)homology in other disciplines and/or nature I recalled Roger Penrose's “On the cohomology of impossible figures": http://upcommons.upc.edu/revistes/...
- 16.5k
12
votes
0
answers
580
views
Cohomology and conifold transition for the quintic
Let $Y\subset \mathbb{C}P^4$ be the quintic threefold given by the equation $$X^5_0+X^5_1+X^5_2+X^5_3+X^5_4+5X_0X_1X_2X_3X_4=0$$
it has 125 singular points whose links are homeomorphic to $S^2\times S^...
- 9,870
12
votes
0
answers
1k
views
Eilenberg-Steenrod axioms of sheaf cohomology
Cohomology of a space is often defined axiomatically: a cohomology theory is a functor
from pairs of spaces to abelian groups
satisfying the Eilenberg-Steenrod axioms. Is there a similar ...
- 40.2k
12
votes
0
answers
1k
views
Are there exotic $S^2\times S^2$?
On 2010 AKHMEDOV and PARK claimed there are infinitely many exotic smooth structures on $S^2\times S^2$, see http://arxiv.org/abs/1005.3346
Then Rasmussen posted a paper : Perfect Morse functions and ...
- 2,623
12
votes
0
answers
434
views
Higher holonomies for higher local systems
In Jacob Lurie's classification of tqfts, one finds a version of the cobordism hypothesis for $(X,\zeta)$-structure, where an $(X,\zeta)$ structure on a manifold $M$ is the datum of a continuous map $...
- 6,777
12
votes
0
answers
444
views
Nullstellensatz for quaternionic plane curves?
By a quaternionic plane curve I mean the zero locus of a noncommutative polynomial in two variables, $x$ and $y$ say, over ${\Bbb H}$, Hamilton's quaternions. It is evidently well-known that, after ...
- 17.6k
12
votes
0
answers
470
views
What is the history of the notion of subdivision of categories?
A recent answer by Peter May prompts me to ask a question which I have been considering to ask for several months. (The reason why I have not asked it before is that it is not directly related to my ...
- 2,413
12
votes
0
answers
414
views
Hilton-Eckmann dual of the Steenrod Algebra
In essence my question can be stated as follows: fill in the analogy
$$
\text{cup product} \qquad\qquad \leftrightarrow \qquad \text{Samelson product}
$$
$$
\updownarrow \qquad\qquad \...
- 18.9k
12
votes
0
answers
661
views
Mapping cylinders of fibrations
If $p: E\to B$ is a fibration, is the map $q:M_p \to B$ from the mapping cylinder
of $p$ also a fibration?
I know that it is if $p$ is trivial, or locally trivial; and I know (from Strøm's "The ...
- 12.5k
12
votes
0
answers
440
views
K-Weil cohomology theories?
I don't know very much about this stuff, so I'm a bit afraid that I'm being naive or stupid, and I apologize if I am --- but it seems to me that Weil cohomology theories, or at least the standard ...
- 21k
11
votes
0
answers
225
views
The algebras and coalgebras of the homology functor
My question is very simple, but I suspect far from the intuition with which singular homology is introduced.
Consider singular homology as a functor
$$H_n : {\sf Top}\times{\sf Ab} \to \sf Ab$$
This ...
- 13.6k
11
votes
0
answers
221
views
On an Artin (?) subgroup of braid groups
While working on something apparently unrelated I encountered a "braid-like" group, which is a relatively geometric subgroup of a braid group and seems to be itself an Artin group. It seems ...
- 42.4k
11
votes
0
answers
336
views
Lattices and stable homotopy groups of spheres
The number $65520$ arises in two very different scenarios:
It occurs in the formula for the theta series of the Leech lattice:
$$ \Theta_{\Lambda_{24}}(q) = 1 + \sum\limits_{m=1}^{\infty} \dfrac{...
- 12.4k
11
votes
0
answers
211
views
Product on cellular cochains of the real Grassmannian
The real Grassmannian $Gr(k,n)$ of $k$-planes in $\Bbb R^n$ admits a Schubert cell decomposition, with one cell for each Young diagram $\lambda$ of height $\leq k$ and width $\leq (n-k)$; the ...
- 52.7k
11
votes
0
answers
335
views
Isotopy on embedded 3-manifolds in 4-manifolds
Suppose that $X$ be an oriented, closed four-manifold. Suppose that $Y$ is an oriented, closed three-manifold smoothly embedded in $X$. Suppose also that $f:X \to X$ is a diffeomorphism that fixes $Y$...
- 3,751
11
votes
0
answers
312
views
Is there a homotopy coherent analogue of Dieudonné modules?
Let $K$ be a perfect field with characteristic $>0$ and $\mathcal{H}$ the category of graded connected abelian hopf algebras over $K$.
By a theorem of Schoeller there is a canonical equivalence ...
- 1,361
11
votes
0
answers
636
views
Does quantum cohomology have an $E_\infty$-ring structure?
Consdier the classical singular cohomology ring $H^*(X, \mathbb{Z})$ of a manifold $X$. The $H^n(X, \mathbb{Z})$'s are the cohomology groups of a chain complex and it turns out that the multiplication ...
- 111
11
votes
0
answers
879
views
Infinite-dimensional affine space in algebraic geometry and algebraic topology
In topology, there are of course many different infinite-dimensional topological vector spaces over $\mathbb R$ or $\mathbb C$. Luckily, in algebraic topology, one rarely needs to worry too much about ...
- 64k
11
votes
0
answers
533
views
Chromatic Homotopy Theory and Physics
Chromatic homotopy theory is a subfield of stable homotopy theory that studies complex-oriented cohomology theories from the "chromatic" point of view, which is based on Quillen's work relating ...
- 10.5k
11
votes
0
answers
332
views
$\Gamma$-sets vs $\Gamma$-spaces
I know that every $\Gamma$-space is stably equivalent to a discrete $\Gamma$-space, i.e. a $\Gamma$-set.
For example, Pirashvili proves, as theorem 1.2 of Dold-Kan Type Theorem for $\Gamma$-Groups, ...
- 42.4k
11
votes
0
answers
266
views
Madsen-Tillmann spectrum $MTE$ of the group $E$ which is defined in Freed-Hopkins's paper
In Freed-Hopkins's paper, the group $E(d)$ is defined to be the subgroup of $O(d)\times\mathbb{Z}_4$ consisting of the pairs $(A,j)$ such that $\det A=j^2$, where $\mathbb{Z}_4=\{\pm1,\pm\sqrt{-1}\}$ ...
- 1,329
11
votes
0
answers
331
views
If an additive group of $\Bbb R^2$ contains a smoothly deformed circle, is it necessarily all of $\Bbb R^2$?
It can be shown that if an additive subgroup of $\Bbb R^2$ contains the unit circle, then it is necessarily all of $\Bbb R^2$. Does this also hold for a suitably smoothly deformed unit circle?
...
- 2,079
11
votes
0
answers
474
views
Is the $\hat{A}$-genus invariant under crepant birational maps between smooth algebraic varieties?
The degree of the Todd class of the tangent bundle of a variety $Y$ gives the holomorphic genus of $Y$, which is a birational invariant. We also know that the $\hat{A}$-roof of a smooth variety $Y$ ...
- 3,022
11
votes
0
answers
264
views
Direct proof of the equivalence of symmetric monoidal $K$-theory and exact sequence $K$-theory?
When all exact sequences split in $C$, we have $\Omega B C \simeq K(C):=\Omega Q(C)$. Heuristically, this is because the space of upper-triangular matrices is contractible. Can this be made precise? I ...
- 64k
11
votes
0
answers
656
views
What is known about mapping class groups of 4-manifolds?
I am mostly interested in the case when you have a smooth degree $d$ algebraic surface $X$ over $\mathbb C$ and we can define three distinct groups: $\pi_0(\mathrm{Diff}^+(X))$, $\pi_0(\mathrm{Homeo}^+...
- 765
11
votes
0
answers
650
views
Triangulation of manifolds with corners
Let's begin with some definitions:
A (smooth) manifold with corners is a Hausdroff (and second countable if you want) space that can be covered by open sets homeomorphic to $\mathbb R^{n-m} \times \...
- 909
11
votes
0
answers
532
views
Third cohomology of symplectic $6$-manifolds
Suppose that $A$ is a finitely generated abelian group with even rank and without $2$-torsion. Does there exist a compact symplectic $6$-manifold $(M,\omega)$ such that $A$ Is isormorphic to $H^{3}(M,\...
- 6,995
11
votes
0
answers
340
views
$K$-theory spectrum of the category of finite groups
(I asked some people this question in person and got the answer "no", but wanted to see if the Internet had more to say)$
\newcommand{\FinGrp}{\mathbf{FinGrp}}
$
Way back in my first group theory ...
- 1,838
11
votes
0
answers
450
views
$E_\infty\mathrm{Spaces}(\mathbf{Z}/p\mathbf{Z},GL_1(E_n))$ and Eilenberg-Maclane spaces
$\newcommand{\Z}{\mathbf{Z}}$Let $p$ be a prime. In his answer here, Jacob Lurie conjectured that $E_\infty\mathrm{Spaces}(\mathbf{Z}/p\mathbf{Z},GL_1(E_n))\simeq K(\Z/p\Z,n)$ where $E_n$ denotes the ...
- 5,760
11
votes
0
answers
372
views
MU and the integral Hodge Conjecture
Totaro proved that the cycle class map for an algebraic variety factors through complex cobordism. In other words, we have a factorization
$$CH^i(X) \rightarrow MU^{2i}(X) \rightarrow H^{2i}(X; \...
- 2,809
11
votes
0
answers
1k
views
Equivariant and orbifold Chern classes
Edit. After thinking about this problem a bit longer, I am not so sure anymore that the Bredon cohomology proposed by Adem and Ruan gives me the invariants I am looking for. I have therefore moved ...
- 6,813
11
votes
0
answers
241
views
Cohomology of a configuration space of points on $\mathbb C^\times$ with an additional restriction
Let $Conf_{1,n}^3$ be the configuration space of collections of $n$ distinct numbered points on the annulus $\mathbb C^\times$ with an imposed restriction: for any $r\in \mathbb R^+$ the circle $\...
- 401
11
votes
0
answers
402
views
How does the HHR Norm functor interact with the cotensor over $G$-spaces?
Let $N_H^G$ be the norm functor from orthogonal $H$-spectra to orthogonal $G$-spectra. We know the category of orthogonal $G$-spectra $\mathcal{S}_G$ is enriched over the category of based $G$-spaces $...
- 423
11
votes
0
answers
669
views
Pairing of cohomology and homology Künneth formulas
Let $k$ be a field, and let $X$ and $Y$ be CW-complexes of finite type (although the question makes sense for $k$ a ring and for more general chain complexes of finitely generated free abelian groups)....
- 35.9k
11
votes
0
answers
2k
views
Total spaces of tangent/cotangent bundles in a course where all varieties are quasi-projective
$\def\PP{\mathbb{P}}$In a course where all varieties are quasi-projective (as in Shafarevich Volume I), I am trying to figure out whether I can justify talking about the total spaces of the tangent ...
- 156k
11
votes
0
answers
292
views
Goodwillie calculus and morphisms of functors
Let $F,G: \mathcal{T}\to \mathcal{S}$ be two functors from topological spaces to spectra (or topological spaces) and let $s: F\to G$ be a morphism between them.
Suppose $F$ and $G$ are analytic and ...
- 743
11
votes
0
answers
648
views
Fields in Stable Homotopy Theory
It is known that the only "fields" in stable homotopy theory, after localizing at a prime $p$, are Eilenberg-Mac Lane spectra for fields and the Morava K-theories (this is true in a few senses: these ...
- 10.5k