All Questions
2,364 questions with no upvoted or accepted answers
8
votes
0
answers
173
views
Stratification of space of labelled circles in the plane
Consider the space of $n$ round circles in the plane to be the open subset of $\mathbb R^{3n}$:
$$C_n = \{ (v_1, v_2, \cdots, v_n, r_1, r_2, \cdots, r_n ) : v_i \in \mathbb R^2, r_i \in (0, \infty) \ ...
- 44.4k
8
votes
0
answers
171
views
Local formula for the signature of $4k$-manifold
In "The Euler characteristic is the unique locally determined homotopy invariant of finite complexes" on page 61 in the penultimate paragraph, Levitt mentions that if one restricts to compact PL $4k$-...
- 5,349
8
votes
0
answers
819
views
Second homotopy group of a topological group
It is well-known that any Lie group $G$ has $\pi_2(G)=0$: see this question. Is the same true for any compact (Hausdorff) topological group? Or even for locally compact ones? Maybe there is a way of ...
- 3,146
8
votes
0
answers
321
views
When does p-profinite completion commutes with maps from a $p$-finite space?
background
Let $\mathcal{S}$ be the ($\infty$-)category of spaces and $\mathcal{S}_{p-\text{finite}}$ the full subcategory spanned by the $p$-finite spaces (that is, the spaces with finitely many ...
- 2,320
8
votes
0
answers
281
views
Combinatorial spin$^{\mathbf{C}}$ structures
Below is a brief introduction to spin$^{\mathbf{C}}$ structure that I took from Wikipedia. For more information, one should refer to https://en.wikipedia.org/wiki/Spin_structure#SpinC_structures.
A ...
- 875
8
votes
0
answers
365
views
$C_2$-equivariant Betti realization of MGL
Let $MGL$ denote the motivic spectrum representing algebraic cobordism. Over $\mathop{Spec}(\mathbb{C})$ there is a Betti realization functor $\mathop{SH}(\mathbb{R}) \to \mathop{SH}$, which takes $...
- 3,784
8
votes
0
answers
134
views
Rational homotopy type of Hilbert scheme components
What is known about rational homotopy type of irreducible component of $Hilb^n(\mathbb C^k)$ containing configuration space? I've searched arXiv for a while and found nothing but surmise that Betti ...
- 4,600
8
votes
0
answers
292
views
Loop space functor and sequential colimits of inclusions
The question is about a fact that is mentioned as "evident" everywhere in the literature, so my guess is that some small detail is passing over my head. Here it is:
Let $X_0\hookrightarrow X_1 \...
- 91
8
votes
0
answers
290
views
Geometric meaning of Aomoto complex
Generally, if we have any space $X$, then multiplication by any odd class $\eta$ makes $H^*(X)$ a complex (usually called Aomoto complex) $A_{\eta}$ because cup product is graded commutative. It is ...
- 4,600
8
votes
0
answers
318
views
Relationships between the field of definition of an algebraic variety and its topology
Let $V$ be a smooth complex projective variety, I was wondering if there exists some homotopical or topological necessary conditions for $V$ to be deformation equivalent to a variety $V_0$ defined ...
- 9,870
8
votes
0
answers
242
views
Framed higher Hochschild cohomology
Given an $E_n$-algebra $A$, one can define its $E_n$-Hochschild complex $CH_{E_n}(A,A)$ by the formula $$Ch_{E_n}(A,A)=RHom_{Mod_A^{E_n}}(A,A)$$ where $Mod_A^{E_n}$ is the category of $A$-modules over ...
- 1,609
8
votes
0
answers
359
views
Tornehave's preprint "On BSG and the symmetric groups"
There are a few papers that cite Tornehave's preprint entitled On BSG and the symmetric groups apparently dating from early 70s or late 60s. Google search reveals very little. Does anyone have access ...
- 7,656
8
votes
0
answers
494
views
Two pictures of K-theory and Bott periodicity
Let me recall the definition of the Bott periodicity isomorphism in the context of $C^*$-algebras. We take a (class of) projection $p \in M_n(A^+)$ and map it to the class of $M_n(A)$ valued loop $f_p$...
- 9,340
8
votes
0
answers
265
views
What is known about maps between loop spaces of Spheres? - Reference request
What is know in general about the maps $\Omega^rS^n\rightarrow\Omega^sS^m$ between loop spaces of Spheres, or, perhaps to phrase it better, the groups $[\Omega^rS^n,\Omega^sS^m]$ for various values ...
- 5,596
8
votes
0
answers
736
views
Homology of inverse limits over inverse systems more complicated than towers
Most textbooks discussions of homology of inverse limits of chain complexes consider only “towers,” i.e. inverse systems indexed by the natural numbers. I’d like to find a reference that explains what ...
- 5,544
8
votes
0
answers
234
views
Explicit diffeomorphism between an infinite dimensional sphere its product with itself
Let $S$ be an infinite dimensional sphere in a Hillbert space.
As $S$ is homotopic to the product $S \times S$, then $S$ is diffeomorphic to $S \times S$ (for Hilbert manifolds, a homotopy ...
- 606
8
votes
0
answers
231
views
Are there genera for algebraic cobordism?
For real and complex manifolds, we can form the (oriented) cobordism ring $\Omega$, and a genus is defined to be a ring homomorphism
$$\varphi:\Omega\otimes\mathbb{Q}\to R$$
where $R$ is any ...
- 3,799
8
votes
0
answers
342
views
Reference request: Whitehead product and the Borel construction
This is a question about signs.
Fix
a based space $(X,x_0)$,
a topological group $G$
acting on $X$ from the left, so that the basepoint $x_0$ is fixed,
a based map $\alpha\colon S^p\to G$ ($p\geq1$)...
- 27.2k
8
votes
0
answers
416
views
Pedagogical question on Lie groups vs. matrix Lie groups
There are two common approaches taken in introductory texts on Lie groups: studying all Lie groups, or focusing only on matrix Lie groups. The main advantage of the latter approach is that one can ...
- 28.1k
8
votes
0
answers
352
views
Two proofs of the Cheeger-Müller theorem
In the late 1970's, Cheeger and Müller independently proved the equality of analytic torsion and Reidemeister torsion for orthogonal representations, which had been conjectured by Ray-Singer. Their ...
- 266
8
votes
0
answers
278
views
Which map realizes the isomorphism $KO_n(X)\otimes \mathbb{Q}\to \bigoplus_{i\in\mathbb{Z}}H_{n-4i}(X;\mathbb{Q})$?
The description of the real $KO$-homology groups $KO_n(X)$ can be given abstractly as maps to the real K-theory spectrum $KO$ smash $X$, or via triples $(M,x,\phi)$ where $M$ is a closed manifold, $\...
- 362
8
votes
0
answers
756
views
Algebraic geometry introduction for homotopy theorists/algebraic topologists
Algebraic geometry has a plenty of decent introductory texts now. Some are of the classical commutative algebraic approach(following EGA), like Ravi Vakil's "Foundations".
Some use facts from ...
- 81
8
votes
0
answers
222
views
Geometric argument for "easy" part of Jordan-Brouwer separation theorem without local flatness
Let $M^n \subset \mathbb{R}^{n+1}$ be an $n$-dimensional compact connected topological submanifold. The Jordan-Brouwer separation theorem says that $\mathbb{R}^{n+1} \setminus M^n$ contains two ...
- 321
8
votes
0
answers
171
views
Which -icial sets produce the "standard" representations of symmetric groups?
Suppose you have a system of cell complexes (say, even convex polyhedra) $(P_n)_{n\geqslant0}$ which occur as faces of each other and are used to define the corresponding notion of "$P_*$-set". So ...
- 18.4k
8
votes
0
answers
193
views
Can stable stems be generated by homotopy operations?
The motivation for this question comes from J. Cohen's result; at the prime $p=2$ his result says that any element in ${_2\pi_*^s}$ can be written as a (higher) Toda bracket of $2,\eta,\nu,\sigma$, ...
- 3,173
8
votes
0
answers
316
views
A model category for E-infty algebras in a non-monoidal model category?
Given a suitable nice symmetric monoidal category $C$, symmetric monoidally enriched, tensored, and cotensored over a symmetric monoidal category $S$, and an operad $\mathcal{O}$ in $S$, we can ...
- 27.5k
8
votes
0
answers
500
views
Failure of "equivariant triangulation" for finite complexes equipped with a $G$-action
Let $\mathcal{S}$ be the $\infty$-category of spaces, and let $G$ be a finite group, and let $BG$ be the groupoid with one object and automorphisms given by $G$.
Consider the $\infty$-category $\...
- 25.6k
8
votes
0
answers
390
views
Classifying space of the higher-structure diffeomorphism group
There is a higher extension of the classifying space $B \mathrm{Diff}$ of the diffeomorphism group implicit in the (infinity,n)-category of cobordisms with (X,zeta)-structure $\mathrm{Bord}_n^{(X,\...
- 19.9k
8
votes
0
answers
554
views
Lower semicontinuity of naive fiber size
I would like to present the following result in my algebraic geometry class, but it is seeming much harder than I would expect. Since my class is working with closed points over an algebraically ...
- 156k
8
votes
0
answers
381
views
Reference for maps whose pushouts are also homotopy pushouts
Consider a category C with weak equivalences, e.g., a model category.
For the purposes of this question, let's say that a morphism f in C is an i-cofibration if any pushout (alias cobase change) ...
- 37.8k
8
votes
0
answers
463
views
On the cohomology of Kontsevich graph complex
Kontsevich's formality theorem asserts that a certain quasi-isomorphism of chain complexes between the graded Lie algebra of polyvector fields on $\mathbb{R}^n$ and the dg Lie algebra of ...
- 1,609
8
votes
0
answers
294
views
Picard-Brauer exact sequence for infinity categories
This question may be very naive, or the answer may be well-known. In any case, a good amount of googling did not bring up anything useful (maybe I'm using the wrong words?).
If $f:A\to B$ is a ...
- 1,961
8
votes
0
answers
180
views
$v_1$-periodic homotopy and principal bundle classification
This question came back to my mind while pondering this MO question. The classification of principal bundles is seriously difficult because of our lack of understanding of homotopy groups of compact ...
- 17.4k
8
votes
0
answers
4k
views
Kunneth spectral sequence
In Rotman's Homological Algebra, 1st edition, there is written:
Is every detail of 11.31-11.35 correct? Isn't the spectral sequence in 11.35 1st quadrant and not 3rd quadrant? Do 11.34-35 also hold ...
- 1,589
8
votes
0
answers
337
views
What's the Hochschild homology of the category of constructible sheaves?
Let $X$ be a manifold. Does the Hochschild homology/cohomology of the category of constructible sheaves on $X$ have a more familiar name?
- 8,723
8
votes
0
answers
550
views
Homology of compact symmetric spaces
Could anyone point me to a table giving the homology of all compact symmetric spaces, i.e. $G/U$ where $G$ is a compact Lie group and $U$ the fixed points of an involution of $G$? I'd be happy even ...
- 276
8
votes
0
answers
213
views
A classification of smooth $S^1$-actions on $\mathbb CP^3$?
Question 1. Is there a classification of smooth $S^1$-actions with isolated fixed points on the standard $\mathbb CP^3$?
Question 2. What if one additionally imposes the condition that the action ...
- 14.3k
8
votes
0
answers
285
views
Systoles of hyperbolic (Riemann) surfaces of large genus
Let $m$ be a Riemannian metric on $S_g$ the surface of genus $g$, and $sys(m)$ be the length of the shortest non contractible cycle with respect to $m$.
The systolic inequality claims that for any ...
- 1,303
8
votes
0
answers
211
views
Fibrations of orthogonal G-spectra and fixed points
There are at least two fixed point functors that characterize stable equivalences of orthogonal G-spectra: the geometric fixed points and the naive fixed points of a fibrant replacement.
Is this true ...
- 425
8
votes
0
answers
813
views
looping and delooping spaces and categories
I'm trying to understand the relationship between the notions of looping and delooping in category theory and topology.
The morphisms in a category with one object have the structure of a monoid. ...
- 1,446
8
votes
0
answers
371
views
Two ways of getting a cohomology class from an extension of a discrete group by $\mathbb C^*$
Suppose that $\overline G$ is a Lie group such that the connected component of $1$ is $\mathbb C^*$. Assume that $\mathbb C^*$ is central in $\overline{G}$, and set $G := \overline G/\mathbb C^*$. ...
- 27k
8
votes
0
answers
636
views
Chern Classes of Exterior Products of a vector bundle.
This is mostly a question in combinatorics. Is there a clean way in terms of determinantal identities to write down $c(\wedge^k V)$ i.e. the individual summands in terms of the individual summands of $...
- 327
8
votes
0
answers
549
views
Description of virtual fundamental class
For some concrete examples, is there an easy way to describe the virtual fundamental class (say, by capping off the moduli pace with an obstruction bundle ). Consider the moduli space of stable maps ...
- 147
8
votes
0
answers
373
views
Completion of n-fold Segal spaces
During a reading course about Jacob Lurie's paper about Local TQFTs, I came across the notion of Segal spaces as models for $(\infty,1)$-categories. Looking at things like the "double nerve" for 2-...
- 7,637
8
votes
0
answers
706
views
Finite generation of equivariant cohomology rings
Let $G$ be a finite group acting on a topological space $X$. (In my applications, $X$ is the classifying space of a compact Lie group.) Let $H^\ast(-)=H^\ast(-;\mathbb{Z}/2\mathbb{Z})$ denote mod 2 ...
- 35.9k
8
votes
0
answers
370
views
Dualizing complex of the product of two locally compact spaces
Hello!
In the setting of locally compact Hausdorff spaces, I would like to understand the relation between the exterior product ${\mathbb D}_X\boxtimes{\mathbb D}_Y$ of the dualizing complexes of two ...
- 2,756
8
votes
0
answers
205
views
Characteristic classes from moduli of alternating forms
Suppose, just for example, that you have a smooth manifold $M$ of dimension greater than $8$,and a cohomology class $q$ in $H^3(M,{\Bbb R})$. Suppose further that one can represent $q$ with a ...
- 17.6k
8
votes
0
answers
340
views
Do p-compact groups have a sufficiently good notion of "flag variety" and "intersection cohomology"?
This is mostly an idle question, since I don't think I'd be able to do anything with a positive answer. But a positive answer would still be interesting, I think.
Background
An outstanding problem ...
- 118k
8
votes
0
answers
438
views
Differential K-theory computation
I am trying to read about K-theory and differential K-theory. I understand that the K-theory of spheres can be computed explicitly (by getting to the stable range and using Bott periodicity and so on)....
- 3,383
8
votes
1
answer
222
views
Non-additivity of intersection forms
Given two oriented $4k$-manifolds $X_1$ and $X_2$, Novikov additivity tells us that
$$
\sigma(X_1 \sharp X_2) = \sigma(X_1) + \sigma(X_2).$$
More generally, if we glue the boundaries of two such ...
- 5,349