All Questions
2,364 questions with no upvoted or accepted answers
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semi flat connections
Let $L\to V$ be complex line bundle and $F_{t}:V\to V$, $t\in [0,1]$, be a loop of diffeomorphisms, $F_0=F_1=$ identity.
For every $x\in V$, we get a loop $\gamma_x(t)=\{F_t(x)\}$ whose class in $\...
7
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192
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Eulerian posets and order complexes
To every poset $P$ it is possible to associate its order complex $\Delta(P)$. The faces of $\Delta(P)$ correspond to chains of elements in $P$. An Eulerian poset is a graded poset such that all of its ...
- 1,889
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219
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Twisting cochain intuition
I'm currently reading through Ed Brown's paper "Twisted tensor products, I", (MR105687, Zbl 0199.58201) and I couldn't find any simple examples of twisting cochains. I understand all ...
- 171
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237
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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
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380
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Reference request: cohomology of BTOP with mod $2^m$ coefficients
I am searching for a reference with information pertaining to the $\mathbb{Z}/{2^m}$ cohomology of ${\rm{BTOP}}(n)$, for $n \geq 8$ and $m=1,2$, where
$${\rm{TOP}}(n) = \{f \colon \mathbb{R}^n \to \...
- 371
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194
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Are the spaces BG for compact connected groups G ind-projective or ind-Kähler?
Let $G$ be a compact connected group, or maybe better its complexification. By thinking about the simplicial Borel space, or using $n$-acyclic $G$-spaces for higher and higher $n$, it's "easy&...
- 44.7k
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191
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Complex cobordism and integrable systems
In Jack Morava's paper On the complex cobordism ring as a Fock representation, it was remarked right at the beginning that complex cobordism may play a role in the theory of integrable systems. In ...
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7
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254
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$C^0$-limit of volume-preserving maps on $\mathbb R^n$
Let $f_k:B_1\rightarrow \mathbb R^n$ be a sequence of injective differentiable volume-preserving maps (i.e. $\mu(f_k(A))=\mu(A)$ for any measurable $A\subset B_1$) that converges uniformly to $f:B_1\...
- 435
7
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170
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Real Bott periodicity in style of Harris’s proof of complex Bott periodicity
Bruno Harris has a beautiful short proof of complex Bott periodicity using the group completion theorem in his paper
B. Harris, Bott periodicity via simplicial spaces. J. Algebra 62 (1980), no. 2, 450-...
- 71
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185
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How does nullification of $K(\mathbb{Z}, 2)$ compare to 1-truncation?
Let $B$ be a type (or space). A type (or space) is $B$-null if the canonical map $X \to X^B$ is an equivalence. The $B$-nullification of an arbitrary type $X$ is a $B$-null type $\bigcirc_B X$ ...
- 4,378
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162
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Relative version of Browder's theorem on H-spaces
A theorem of W. Browder [1] from 1961 says that an H-space $X$ with finitely generated integral homology (meaning that $H_*(X)$ is finitely generated) has $\pi_2(X) = 0$. This generalizes Cartan's ...
- 19.4k
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162
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Poincaré series of deloopings of finite complexes
Recall that the Poincaré series of a topological space $X$ is defined as $P_X(t) = \sum_{j=0}^{\infty} b_jt^j$, where $b_j = \text{dim}_{\mathbb Q} H_{j}(X;\mathbb Q)$ means the $j^{\text{th}}$ (...
- 11.9k
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315
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Can every finitely presented group be realized as a fundamental group of a compact four-dimensional smooth submanifold $\mathbb{R}^4$?
Every finitely presented group is realized as a fundamental group of a two-dimensional complex (a simple exercise on Van Kampen's theorem). I was told that a two-dimensional complex can be well ...
- 6,962
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265
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Homotopy type of space of embeddings of a disk
Let $M^n$ be a smooth $n$-dimensional manifold and let $\mathbb{D}^n$ be the open unit disk in $\mathbb{R}^n$. Consider the space $\text{Emb}(\mathbb{D}^n,M^n)$ of smooth embeddings of $\mathbb{D}^n$ ...
- 44.8k
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comparison of polynomial loop group and smooth loop group
I have a question about Section 8.6 of Pressley-Segal's Loop groups book. Let $G$ be a compact, connected Lie group. Proposition 8.6.6 concerns the comparison of homotopy type between its polynomial ...
- 959
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439
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Transfer of E-infinity algebra structures
Skip to the bottom for my questions, first some discussion:
It is a celebrated theorem of Kadeišvili that $A_{\infty}$-algebra structures can be transferred along homotopy equivalences so that the ...
- 561
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192
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Higher homotopy groups of an orbifold
Given an orbifold $\mathcal{O}$, I have seen many ways to define the orbifold fundamental group:
Thinking of $\mathcal{O}$ as a groupoid $\mathcal{G}$, $\pi_1^{orb}(\mathcal{O})$ can be defined as ...
- 1,452
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302
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The role of spectral Lie algebras and twisting for operads in spectra
In the theory of differential graded (co)operads, the notion of twisting is ubiquitous. The fundamental notion is the twisting map from a cooperad $C$ to an operad $P$. It is defined as a Maurer-...
- 5,849
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324
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Topological operations corresponding to abelianization of the fundamental group
$\newcommand{\ab}{\mathrm{ab}}$Suppose we have a topological space $X$ with a non-abelian fundamental group $\pi_1(X)$. We'd like to perform a sequence of topological operations on $X$ (for example, ...
- 79
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240
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What is the group completion of the groupoid of even finite sets and even permutations?
$\newcommand{\sgn}{\mathrm{sgn}}\newcommand{\defeq}{\overset{\mathrm{def}}{=}}$The Barratt–Priddy–Quillen–Segal theorem says that the $\mathbb{E}_\infty$-group completion of the groupoid of finite ...
- 11.8k
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230
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The homotopy monoids of directed spheres
In directed homotopy theory, one replaces spheres by directed spheres and homotopy groups by homotopy monoids.
Is it known what are the first few homotopy monoids of directed spheres?
Do homotopy ...
- 11.8k
7
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181
views
In what sense do the real and complex places correspond to setting q equal to 1 or -1?
It often happens that if we have a scheme $X/\mathbb Z$ (or an open subset thereof) and we denote by $p(q) = X(\mathbb F_q)$, then $p(1)$ and $p(-1)$ compute the euler characteristic of $X(\mathbb C)$ ...
- 7,746
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131
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Endofunctors of the surface category
Let $\mathrm{Cob}_2$ be the symmetric monoidal $(\infty,1)$-category whose objects are closed oriented $1$-manifolds and whose morphisms are compact oriented surface bordisms. (Higher morphisms are ...
- 1,082
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706
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Update on "A Mad day's work" by Cartier
In his paper "A mad day's work: from Grothendieck to Connes and Kontsevich. The evolution of concepts of space and symmetry," Cartier discusses in the last sections 8 and 9 the role of ...
- 10.5k
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194
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"Relative Whitehead products"
The notion of a relative Whitehead product exists in the literature and has been asked about before (e.g. here). I am trying to find out about a different product on relative homotopy classes which ...
- 208
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152
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Weak homotopy type of the Cech Nerve
Let ${\cal U} = \{U_i\}_{i\in J}$ be an open cover of a topological space $X$, where the indexing set $J$ is assumed to be well-ordered. Then the Cech nerve is the "simplicial space without ...
- 18.9k
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183
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Whitney sum via Gysin
Let $E_1\to E\to E_2$ be a short exact sequence of vector bundles. The Whitney sum formula says that $e(E)=e(E_1)e(E_2)$, i.e. that the Euler class is multiplicative.
Is there a proof of this fact ...
- 5,711
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408
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Low-Dimensional Spaces with High-Dimensional Homology
Barratt-Milnor Spheres $X_n$ are spaces with finite topological dimension $n$ but which have non-vanishing singular homology in arbitrarily high dimensions. Here, they prove that if $n > 1$ then ...
- 439
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350
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Do smooth maps with nowhere-maximal rank have small image?
I’m trying to better understand the concept of “maps with small image” as used by Lipyanskiy in his construction of “geometric homology” in https://arxiv.org/abs/1409.1121. Lipyanskiy utilizes ...
- 5,544
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270
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Differentials in spectral sequences and Massey products
Consider a multiplicative spectral sequence such as the cohomological Serre spectral. It is known that differentials will satisfy a Leibniz rule. Is there a clean statement involving differentials and ...
- 965
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161
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Homotopy types of causal / chronological pathspaces in Lorentzian manifolds?
Let $M$ be a Lorentzian manifold, and let $p,q \in M$. Let $\Pi^J(p,q)$ be the space of causal paths from $p$ to $q$ (in the compact-open topology).
Question 1: Is it reasonable to expect that the ...
- 64k
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Is there a Swan-style description of topological K-homology?
A celebrated result of Swan [1] states that, on a compact Hausdorff space $X$, the category of finite rank complex vector bundles is equivalent to the category of finitely generated projective $\...
- 1,020
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223
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Does the Whitehead torsion of a homotopy equivalence depend on the CW structure?
In the (old) literature I've seen referenced the question of whether simple homotopy equivalence is a topological property, i.e. whether it depends only on the underlying space, rather than the ...
- 5,849
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424
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Calculation of $\pi_3^s$ via killing spaces and Steenrod squares
$\newcommand{\Sq}{\text{Sq}}
\newcommand{\Z}{\mathbb{Z}}
\newcommand{\Sf}{\mathbb{S}}$I am grateful to anyone who will help me, because I know it is not a short calculation and it requires some time. ...
- 1,263
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541
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Convergence of a spectral sequence of a double complex
In Weibel's book, a spectral sequence $E^r_{p,q}$ is said to weakly converge to a graded object $H_{\ast}$ if for every $n$ there exists a filtration $\dots \subset F_{r}H_{n} \subset F_{r-1}H_{\ast} \...
- 1,325
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253
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Applications of the Atiyah-Patodi-Singer eta-function $\eta(s)$
The eta function of a differential operator was used by Atiyah, Patodi and Singer to derive their famous index theorem, and is given by
$$
\eta(s)=\sum_{\lambda\neq 0}(\mathrm{sign}\lambda)|\lambda|^...
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363
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What is a morphism of ∞-sites?
Recall that a morphism of sites
is a covering-flat functor
that preserves covering families.
Morphisms of sites can be identified with those
geometric morphisms of induced toposes
for which the ...
- 37.8k
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206
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A confusion about geometric fixed points via spectral Mackey functors and smashing localisations
Let $G$ be a finite group and $N$ a normal subgroup. One of the modern ways to construct the $\infty$-category of $G$-spectra is as product-preserving spectral presheaves $\text{Sp}^G = \text{Fun}^{\...
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Homological and homotopical equivalence of complex analytic varieties
Consider a map between two complex analytic varieties of finite type $f:X\to Y$. Suppose that $f$ induces isomorphisms on cohomology with (constant) integral coefficients. Under what reasonable ...
- 455
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Cohomology with group ring coefficients and compact support
Let $X$ be a contractible space and let $G$ be a group acting freely, properly discontinuously, and cocompactly on $X$. We get an induced action on the singular chain complex $C_{\bullet}(X)$.
...
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Simplicial model for $\mathcal{L}BG//S^1$ for a finite group $G$
$\require{AMScd}$For $X$ a (nice enough) topological space, the free loop space $\mathcal{L}X$ is the space of continuous maps from $S^1$ to $X$. This space has a natural $S^1$ action given by ...
- 6,777
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273
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Generalization of familiar theorem about singular homology to general model category
I have two questions, the first one is just wether the following statement is true or not? Is there a reference for this?
The second question is maybe related, I don't know. But anyway, given $U:\...
- 363
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333
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Positive instances of the Eilenberg-Ganea conjecture with families
The original Eilenberg-Ganea conjecture, which remains unsettled, can be formulated as: any (discrete) group $G$ of cohomological dimension $\operatorname{cd}(G)=2$ has geometric dimension $\...
- 35.9k
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223
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Duality of Hopf algebras and duality of spectra
Let $S$ be the sphere spectrum, and for $X$ a topological space, let $S(X)$ be the mapping spectrum from the free loop spectrum on $X$ to the sphere spectrum. This is an $E_\infty$ ring spectrum (also ...
- 7,962
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121
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Is the deformation along flow lines a simple homotopy equivalence?
Let $(M,g)$ be a compact, smooth $n$-manifold with boundary $\partial M$ and let $f: M \to [a,b]$ be a Morse function, whose critical points are interior and which satisfies $f^{-1}(b) = \partial M$.
...
- 1,255
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172
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Is there an $\infty$-topos of monochromatic spaces?
Fix (a prime $p$ and) a chromatic height $h$. Recall that the Bousfield-Kuhn functor $\Phi_h: \mathcal M_h^f \to Sp_{T(h)}$ is monadic, where $\mathcal M_h^f \subseteq Top_\ast$ is a certain ...
- 64k
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221
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Which spaces are most naturally presented simplicially?
It's often a great technical convenience to know that you can "do homotopy theory" with simplicial sets. But if you really get down to it geometrically, it can be awkward.
In general, if I have a CW ...
- 64k
7
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328
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Funtoriality of twisted K-theory
I posted this question on math.stackexchange, but received no answer there.
In order to avoid the XY problem I will first state what I want, then what I think is the solution and how that failed until ...
- 301
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344
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Reference request: complex K-theory as a commutative ring spectrum
Does anyone know of a point-set level model for complex K-theory as a commutative ring spectrum?
For real $K$-theory
I know of "A symmetric ring spectrum representing KO-theory" by Michael Joachim (...
- 401
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282
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A cohomology associated to a vector field on a Riemannian manifold
Edit: Accoring to the comment of Asura Path I revise the question.
Let $X$ be a vector field on a Riemannian manifold $(M,g)$. So we have a $1$-form $\beta$ with $\beta(Y)=\langle Y,X\...
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