All Questions
2,364 questions with no upvoted or accepted answers
8
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A variation of necklace splitting
Our problem is the following:
Let $n$ and $k$ be integers.
We are given two (unclasped) necklaces, each with $n$ colored stones: a top necklace which has $k$ colors and a bottom necklace which has 2 ...
- 81
8
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156
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Generators for unstable cobordism
I am looking for explicit descriptions of generators of some low-dimensional unstable cobordism groups. For example, $\mathbb CP^2$ embeds into $\mathbb R^7$ by a result of Haefliger. Because it has ...
- 6,813
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206
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Universal bundles for monoids versus groups
Dold and Lashof compare their construction for a monoid M to Milnor's
when M is a group G. They give an explicit comparison for the first stage of the constructions. Somewhere I've seen the general n-...
- 301
8
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172
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The pro-discrete space of quasicomponents of a topological space
Let $X$ be a topological space.
Consider the functor $P^X : \textbf{Set} \to \textbf{Set}$ that sends each set $Y$ to the set of continuous maps $X \to Y$.
It is not hard to check that $P^X : \textbf{...
- 15.9k
8
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360
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Worst-case complexity of calculating homotopy groups of spheres
Is the best known worst-case running time for calculating the homotopy groups of spheres $\pi_n(S^k)$ bounded by a finite tower of exponentials? How high is a tower? Does $O(2^{2^{2^{2^{n+k}}}})$ ...
- 827
8
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202
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Looking for the source of expository passages concerning the Adams spectral sequence
The following question might be off-topic strictly speaking, but it does have the form of a reference request (in some sense) and also I think that enough members of the relevant community are MO ...
- 25.8k
8
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290
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When does the tangent microbundle of a closed orientable topological $4k$-manifold have a trivial rank 2 subbundle?
$\DeclareMathOperator{\Top}{Top}
\DeclareMathOperator{\co}{H}$Let $M$ be a closed orientable connected topological manifold of dimension $4k$ with $k > 1$. It is known (David Frank, On the index of ...
- 1,816
8
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138
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When is an ideal in the cohomology ring the kernel of a map induced by a map of spaces?
Let $X$ be a space and $I$ be an ideal in the cohomology ring $H^*(X)$.
I am interested in the question whether there is a map of spaces $Y\rightarrow X$ such that $I$ is the kernel of the induced map ...
- 11.1k
8
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240
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Linear $S^{2k}$-bundles over $S^{4k}$
By the classification of Dold and Whitney, linear $S^2$-bundles over $S^4$ are classified by their first Pontryagin class $p_1$, which takes the value $4\lambda$ for the bundle corresponding to $\...
- 291
8
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300
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What can I say about an $E_\infty$ ring spectrum with an odd invertible element?
I have an $E_\infty$ ring spectrum $R$. I suspect it is trivial, but I'm not sure. What I do know is that there is an $R$-linear equivalence $R \simeq \Sigma R$. Unless I am very confused, this ...
- 54.6k
8
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441
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Poincaré duality for topological $K$-theory
Let $n$ be an even number. Let $X$ be a $n$-dimensional complex projective manifold with
$H^{2m+1}(X,\mathbb{Z})=0$, for all $0\leq m\leq n-1$.
$H^{2m}(X,\mathbb{Z})$ is a free $\mathbb{Z}$-module ...
user39380
8
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189
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Conner-Floyd Chern classes and $E$-(co)homology of $BU$
In his book, Stable homotopy and generalised homology, Adams computes the $E$-(co)homology of $BU$ for a complex oriented cohomology theory $E$. In II.4, he first describes the $E$-homology of $BU$ as ...
user340793
8
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267
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$\mathbb RP^n$ bundles over the circle, II
EDIT: I fixed the issue pointed out by Nicholas Tholozan, thanks for sheding light on this!
This question is written as a follow-up to this one.
Both answers there are great, but my impression is ...
- 11.9k
8
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223
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Representing the fundamental class of an aspherical manifold in the bar complex
Suppose $M$ is a compact orientable aspherical manifold and $G$ its fundamental group. Is there a nice description of representatives of the fundamental class of $M$ and its dual in the (homogenous) ...
- 1,174
8
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219
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Computation of the 3-dimensional $\mathbb{Z}/m$-equivariant spin cobordism group (with possibly non-empty fixed-point set)?
$\newcommand{\odd}{\mathrm{odd}}\newcommand{\ev}{\mathrm{ev}}$Consider tuples of the form $(Y,\mathfrak{s},\widehat{\sigma})$ where: $Y$ is a closed oriented 3-manifold, $\mathfrak{s}$ is a spin ...
- 299
8
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128
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What are the stable cohomology classes of the "orthogonal groups" of finite abelian groups?
Let $A$ be a finite abelian group, and equip it with a nondegenerate symmetric bilinear form $\langle,\rangle : A \times A \to \mathrm{U}(1)$. Then you can reasonably talk about the "orthogonal ...
- 54.6k
8
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238
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Is there a finite group with nontrivial $H^2$ but vanishing $H^4$, $H^5$, and $H^6$?
Is there a finite group $G$ such that the group cohomology $\mathrm{H}^2_{\mathrm{gp}}(G; \mathbb{Z}/2)$ is nontrivial but $\mathrm{H}^4_{\mathrm{gp}}(G; \mathbb{Z}/2)$, $\mathrm{H}^5_{\mathrm{gp}}(G;...
- 54.6k
8
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299
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(Homotopy) inverse limits of towers of spaces or simplicial sets - reference request
Suppose we have a tower of Kan fibrations between Kan complexes:
$$ X_0 \xleftarrow{f_0} X_1 \xleftarrow{f_1} X_2 \xleftarrow{f_2} \dotsb $$
From this we get a commutative diagram of topological ...
- 56.9k
8
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588
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Values of cohomology theory on a point
$\DeclareMathOperator\Sm{Sm}$It is a well-known fact that in algebraic topology, generalized cohomology theories are determined by their values on the point. I was wondering whether anything similar ...
- 5,901
8
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210
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Is there an example of a compact complex (Kähler) manifold of general type with trivial topological euler charcteristic?
Topological Euler characteristic is given by the top Chern class of the tangent bundle up to sign.
Comments: Examples don't exist in dimension 2 due to Bogomolov–Miyaoka–Yau inequality. Is that true ...
- 452
8
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0
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880
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Nonabelian variants of the Breen-Deligne resolution
The Breen-Deligne resolution is an unusual functorial resolution of an abelian group A by finite direct sums of free abelian groups of the form $\Bbb Z[A^n] = Free_{Ab}(A^n)$. It makes several ...
- 52.7k
8
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201
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Cohomology algebra of the maximal nilpotent subalgebra of a semisimple Lie algebra
The answer to this question may be well-known, but I failed to locate it in any obvious source. From the results of Bott (Ann. Math. 66 (1957), 203-248) and Kostant (Ann. Math. 74 (1961), 329-387), it ...
- 16.9k
8
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251
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(Higher) flat connections and Grothendieck construction
For any (nice) topological space there is an equivalence between covering spaces and local systems $\pi_1X \to \operatorname{Set}$. We can think of it as a Grothendieck construction. If we take ...
- 834
8
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answers
145
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singular homology of manifold with corners
Given two smooth manifolds with corners, let's say that a map $f:X\to Y$ is "transversally smooth" if it is smooth in the usual sense and if (in a local sense on $X$) for every open Whitney ...
- 7,962
8
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681
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Stalks of limit sheaves
Let $\{\mathcal{F}_i\}_{i\in \mathbb{N}}$ be an inverse system of sheaves of abelian groups on a space $X$. Then for any $x\in X$ we have a natural map
$$\left(\lim_i \mathcal{F}_i\right)_x\rightarrow ...
- 4,418
8
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229
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On the homotopy groups of the spectrum $D(n)$
I am interested in learning about the homotopy groups of the spectrum $D(n)$ at the prime $2$ which is defined as the cofibre of the diagonal map
$$Sp^{2^{n-1}}S^0 \to Sp^{2^n}S^0$$
where $Sp$ is the ...
- 3,173
8
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348
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Proving faithful flatness of a K-theoretic map without the moduli stack of formal groups
I'm in the process of writing an expository paper on complex K-theory and Snaith's theorem; the proof of Snaith's theorem that I'm following along (located at http://math.uchicago.edu/~amathew/snaith....
8
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315
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Cohomology of the complement of the resonance hyperplane arrangement
Here was a question about resonance arrangement. It is defined as follows.
Let $x_i$ be the standard coordinates on $\mathbb{C}^n$. For each nonempty $I\subseteq\{1,\dots,n\}$, define the hyperplane $...
- 743
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339
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It is possible that $ X \simeq ΩX $? and that $ X \simeq Ω^ 2X $?
Original post: https://math.stackexchange.com/questions/3810423/it-is-possible-that-x-simeq-%ce%a9x-and-that-x-simeq-%ce%a9-2x
I am studying J. Strom's Modern Classical Homotopy Theory. In chapter 4 ...
- 1,263
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166
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A question on recognition of equivariant loop spaces
I have a question about equivariant loop space that has been bothering me, and that I have not been able to find an answer to in the obvious places.
We know from the work of Segal that to give a loop ...
- 745
8
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217
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Hopf invariants of elements from spherical fibrations
Let $G_n$ be the space of homotopy-equivalences of $S^{n-1}$. Evaluation produces a map $G_{n} \to S^{n-1}$. For $n = 2m+1$, I would like to understand the induced map on $\pi_{4m-1}$. More precisely, ...
- 11.9k
8
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176
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Comparison of two well-known bases of the integral homology group of based loop group
Let $G$ be a compact simply-connected Lie group. Then one can look at the homology $H_*(\Omega G;\mathbb{Z})$ of the based-loop space $\Omega G$ in at least two different ways:
(1) Via Bott-Samelson'...
- 461
8
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0
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869
views
What is known about homotopy groups of spheres?
I'm looking for a list/table/survey of what is known (and what is not known) about homotopy groups of spheres, for example: which are known, which are known stably, which are known primally, non-$0$ ...
8
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394
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Cohomology of constructible sheaves via exit paths
Let $X$ be a stratified space, with stratification $S$ (we will ignore technicalities).
The category of exit paths $Ex(X,S)$ is a directed refinement of the path groupoid of $X$ accounting for the ...
- 1,635
8
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0
answers
332
views
Differential version of $G\mapsto H^3(G,\mathbb Z)$?
Let $\mathit{cLieGrp}^{\mathrm{inj}}$ be the category of compact connected Lie groups, and injective continuous group homomorphisms.
Is there a reasonable functor (some kind of degree $3$ differential ...
- 43.2k
8
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490
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Self diffeomorphism of $S^2\times S^2$
The main question is motivated from the answer of this question https://math.stackexchange.com/questions/2481200/finite-groups-gs-which-acts-freely-on-s2-times-s2
Is it true that every self ...
- 3,751
8
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228
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Does Farjoun's "fiberwise localization" have a universal property?
Let $\mathcal S_L$ be any accessible reflective subcategory of the $\infty$-category of spaces. In his book, Farjoun discusses "fiberwise $L$-localization" of a map of spaces $E \to B$, i.e. a ...
- 64k
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272
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Integral cohomology of compact Lie groups and their classifying spaces
Let $G$ be a compact Lie group and let $BG$ be its classifying space. Let $\gamma\colon \Sigma G \to BG$ be the adjoint map for a natural weak equivalence $G \xrightarrow{\sim} \Omega BG$. Taking ...
8
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319
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Bringing cohomology recipes from algebra to topology?
In algebra, cohomology theories are often defined very roughly like this. Start with a scheme $X$ you want to study. Form a category-with-extra-structure (a site) $\mathcal{C}$ whose objects are, ...
- 1,150
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285
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Are the braid groups good in the sense of Toën?
In this question it is asked whether the mapping class groups are 'good' in relation to pro-finite completion. Helpfully, one of the answers gives a link to a proof that the braid groups are good ...
- 1,635
8
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342
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Lusternik-Schnirelmann Category of 4-Manifolds
Whilst reading 'the book' I stumbled across the following elegant little theorem, due to Gómez-Larrañaga and González-Acuña.
Let $M$ be a closed $3$-dimensional manifold. Then its Lusternik-...
- 5,596
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answers
144
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homotopy MC element
A homotopy MC element is the Linfty analog of a Maurer-Cartan
element for a Lie algebra. Where is anything written about
homotopy MC elements as perturbations of strict MC elements?
- 3,880
8
votes
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answers
204
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Relationship between the p-radical subgroups and the parabolics in a BN-pair generality
A theorem of Quillen says that if $G$ is a finite Chevalley group over characteristic $p$, then the poset $\mathcal{A}_p(G)$ of nontrivial elementary abelian subgroups of $G$ is homotopy equivalent (I ...
- 1,816
8
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272
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Generalize Wu formula to general Bockstein homomorphisms
The classical Wu formula claims that
$$Sq^1(x_{d-1})=w_1(TM)\cup x_{d-1}$$
on a $d$-manifold $M$, where $x_{d-1}\in H^{d-1}(M,\mathbb{Z}_2)$.
I wonder whether there is a generalization of the ...
- 1,329
8
votes
0
answers
125
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Relating bordism generators in d and d+2 dimensions --- an explicit example
This is an attempt to make my relation between bordism invariants in $d$
and $d+2$ dimensions, following a previous attempt more explicit. This counts as a different question, since some more specific ...
- 2,147
8
votes
0
answers
198
views
"Gerbes" in the cobordism theory
In a lecture I attended today, I heard the use of gerbes in the cobordism theory.
Previously, I use cobordism theory, but I never encounter the term "gerbes" when I work on bordism or cobordism group ...
- 10.5k
8
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328
views
Did the Goerss-Hopkins manuscript "Multiplicative stable homotopy theory" ever appear?
A citation to "M. J. Hopkins and P. Goerss, Multiplicative stable homotopy theory, unpublished manuscript, 1996" appears in the Hill, Hopkins, Ravenel Annals paper on the Kervaire invariant. It was ...
- 30.3k
8
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563
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What is $SL(2,\mathbb{R})$-Chern-SImons Theory?
I found in physics that Chern-Simons theory is closely related with three dimensional gravity.
From this paper Three Dimensional Gravity Revisited, the author talks about the Chern-Simons for
$$\...
- 615
8
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0
answers
244
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Clarification of Tillmann's construction of the higher genus surface operad
Sorry if this question is inappropriate for overflow. I tried asking on stackexchange yesterday but didn't get any responses, so I thought that this site might be better. Anyway, my question is as ...
- 383
8
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0
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226
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Are spin Hurwitz numbers $r$-spin Hurwitz numbers?
(I think the answer is no, but I'm not sure.)
In Hurwitz theory, one counts $n$-fold branched covers $\Sigma'\to\Sigma$ of a Riemann surface $\Sigma$ with fixed
ramification data around each branch ...
- 6,881