All Questions
1,240 questions
5
votes
1
answer
246
views
Enriched coends which preserve equivalences
Although this question might be formulated in higher generality, let me try to be concrete:
Let $(\mathbf{Top},\times,*)$ be the monoidal category of compactly generated weak Hausdorff spaces; and let ...
- 1,623
5
votes
1
answer
443
views
endomorphisms of modules over symmetric ring spectra
I have a probably very basic question about modules over symmetric ring spectra:
Let $R$ be a commutative symmetric ring spectrum and let $M$ and $N$ be module spectra over $R$. Moreover, let $\...
- 7,637
5
votes
2
answers
454
views
Burnside ring and zeroth G-equivariant stem for finite G
Let $G$ be a finite group. The theorem that the Burnside ring $A(G)$ is isomorphic to the zeroth stable stem $\pi^{G}_0(S)$ is usually said to originate from Segal. I search for a reference of a proof ...
- 1,273
5
votes
1
answer
825
views
Computing Bredon Cohomology of Z/2-spheres?
Can anyone suggest me how to calculate explicitly the Bredon Cohomology of the sphere(at least for 2-dimensional case) with antipodal Z/2-action with constant coefficient system associated to integers?...
- 745
5
votes
1
answer
549
views
Show that if $p\neq 2$, then $\mathbb{Z}_p$ cannot act freely on $\mathbb{C}P^n$
If $p\neq 2$, then the cyclic group $\mathbb{Z}_p$ has no free continuous action on $\mathbb{C}P^n$. My question is how to prove the above fact using Leray-Serre spectral sequence associated to the ...
- 395
5
votes
1
answer
418
views
$RO(Q)$-graded homotopy fixed point spectral sequence
I am trying to understand some part of J. Greenlees's "Four approaches to cohomology theories with reality": https://arxiv.org/abs/1705.09365
I have a problem with understanding $RO(Q)$-graded ...
- 1,759
5
votes
2
answers
2k
views
Construction of Serre Spectral Sequence
I'm trying to follow Hopkins' construction of the Serre Spectral Sequence, but some "obvious" things are not that obvious to me.
He starts with considering a double complex $C_{\bullet,\bullet}$ with ...
- 305
5
votes
1
answer
277
views
Pontryagin number for 4-dim surface bundle
In paper arXiv:math/0701247
"Divisibility of the stable Miller-Morita-Mumford classes" by Soren Galatius, Ib Madsen, Ulrike Tillmann, it was shown
that the Pontryagin numbers for a 4-dim surface ...
- 4,826
5
votes
0
answers
204
views
homotopy type of box topology.
Suppose that $X$ is weakly equivalent to a point. Let $I$ be a set. Does $\prod_{i\in I}X$ weakly equivalent to a point, where $\prod_{i\in I}X$ is equipped with box topology ?
- 1,613
5
votes
1
answer
388
views
The space of contractible loops of a finite dimensional $K(\pi,1)$
Let $X$ be a finite dimensional $K(\pi,1)$ manifold. Is it true that the space contractible loops of this manifold can be contracted to the space of constant loops on $X$? What if $X$ is a finite ...
- 14.3k
5
votes
2
answers
307
views
3-folds with "simple" Betti numbers and positive Kodaira dimension
I am interested to know an example of a simply connected smooth projective 3-fold $X$ (over $\mathbb{C}$) satisfying the following two constraints:
$X$ has the same Betti numbers as $\mathbb{C}\...
- 6,995
5
votes
1
answer
507
views
Are open orientable 3-manifolds parallelizable via obstruction theory?
In the case of orientable closed $3$-manifolds, we have 4 ingredients that ensure parallelizability:
1a) Closed smooth $n$-manifolds have homotopy type of a $CW$-complex
1b) Closed smooth $n$-...
- 441
5
votes
2
answers
928
views
Stiefel-Whitney classes of a projective space bundle
Hi!
Let $\gamma_1$ denote the twisted line bundle over $S^1$ and add a trivial $(2k-1)$-bundle $\mathbb{R}^{2k-1}$. Consider the projective bundle $P(\gamma_1 \oplus \mathbb{R}^{2k-1})$ over $S^1$. ...
- 81
5
votes
2
answers
728
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Does a manifold which bounds always admit a free involution?
If a closed smooth manifold $M$ admits a smooth free involution $T$, then it bounds. In fact, the mapping cylinder of the quotient map $M \to M/T$ is the manifold whose boundary is $M$.
Is the ...
- 127
5
votes
3
answers
479
views
Is there a dense subset on closed Jordan curve $C$ which its points make intersections under certain rotations?
Is it true that for any given closed Jordan curve of $C \subset \mathbb{R}^2$ there is a dense subset $A$ such that for every point $p\in A$ we have the following property:
If we rotate $C$ around $p$...
- 289
5
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1
answer
287
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Codimension zero embeddings and maps with small fibers
Edit: as explained in my comment on alesia's answer, I mistakenly did not ask below the question I intended (due to my misguided efforts to simplify it). Thus, I revised and reposted my question here.
...
- 501
5
votes
2
answers
562
views
If $E$ maps onto a contractible space with contractible fibers, must $E$ be contractible?
Let $p\colon E\to C$ be a continuous, surjective map between topological spaces with $C$ contractible. Suppose that $p^{-1}(c)$ is contractible for each $c\in C$. Is it true that $E$ is weakly ...
- 393
5
votes
0
answers
359
views
Long exact sequence of Quillen derived functors
Let $F:\mathcal A\to \mathcal B$ be an additive right exact functor between abelian categories $\mathcal A$ and $\mathcal B$. Then for a short exact sequence
$$
0\to A\to B\to C\to 0
$$
in $\mathcal A$...
- 385
5
votes
3
answers
2k
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Classification of simply connected smooth projective varieties?
This question is related to this one.
I am wondering whether there is any sort of classification of simply connected smooth projective varieties, or any work in related directions.
The reason I am ...
- 21k
5
votes
0
answers
171
views
Spectral sequence construction of Euler class of group extension
Let $A$ be an abelian group equipped with an action of a group $G$ and let
$$1 \longrightarrow A \longrightarrow \Gamma \longrightarrow G \longrightarrow 1$$
be an extension of group inducing the ...
- 51
5
votes
1
answer
443
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Classification of fibrations for classifying spaces $B^2\mathbb{Z}_2$ and $BSO(3)$ or $BO(3)$
I am interested in knowing what can we say about the classification of fibrations for classifying spaces $B^2M \equiv B^2\mathbb{Z}_2$ and $BG \equiv BSO(3)$ or $BO(3)$.
Here we can take either:
$B^...
- 2,147
5
votes
2
answers
448
views
computing second cohomology $H^2(O_a,\mathbb{Z})\cong \mathbb{Z}^n$ of a generic coadjoint orbit
Let $G$ be a compact , connected and simply connected Lie group and $a\in\mathfrak{g}^*$ (dual of Lie algebra of Lie group $G$). Then let $O_a$ be a generic coadjoint orbit then can we say $H^2(O_a,\...
user21574
5
votes
1
answer
202
views
Does profinite completion commute with mapping spaces?
Does there exist a prime number $p$ and a smooth complex projective variety $X$ such that $F_{\infty p}\mathrm{Map}(B\mathbb{Z}/p\mathbb{Z}, X)$ is not weakly homotopy equivalent to $\mathrm{Map}(B\...
- 723
5
votes
1
answer
775
views
Cell decomposition for a variety not necessarily complete?
Let $X$ be an algebraic variety with a $\mathbb C^*$ action such that the fixpoints set is finite. By theorem 4.3 in the paper of Bialynicki-Birula "Some theorems on actions of algebraic groups", ...
- 51
4
votes
1
answer
545
views
Is the category of metric spaces and continuous maps Quillen equivalent to Top?
I am looking for models of ${\mathsf{Top}}$ distinct from modifications of simplicial sets. The above question should be understandable to the reader. I'll add more details when I get access to a ...
user2529
4
votes
1
answer
193
views
Version of pseudo-isotopy $\neq$ isotopy for $(n+1)$-framings
Let $M$ be a closed $n$-manifold and $\varphi$ be a self-diffeomorphisms of $M$.
There is a bordism from $M$ to itself given by $M\times [0,1]$ with the identification $M \cong M \times \{0\}$ induced ...
- 890
4
votes
2
answers
921
views
Cone over the Join of two topological spaces
Suppose $X$ and $Y$ are topological spaces. Let's define the join $X\ast Y$ as the quotient space $X\times Y\times [0,1]/\sim$, where $\sim$ is the equivalence relation generated by $(x,y,0)\sim(x,y',...
- 45
4
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0
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181
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Specify the embedding of special unitary group in a Spin group via their representation map
How do we specify the embedding of a Lie group $G_1$ as a subgroup into a larger Lie group $G_2$, with $G_1 \subset G_2$ that agree with a constraint on the mapping between their representations?
By ...
- 10.5k
4
votes
1
answer
595
views
Algorithm for computing fundamental group of simplicial complexes
For computing homology of a simplicial complex, there is the well-known reduction algorithm.
How about for fundamental group of simplicial complexes? Is there any (implementable) algorithm to compute ...
- 1,229
4
votes
1
answer
649
views
Essential simple closed curves on a punctured torus vs those in the torus
Let $T$ be a compact oriented torus, let $p\in T$ be a point, and let $T^*$ be $T - \{p\}$.
In Farb-Margalit's Primer on mapping class groups, in the discussion after Proposition 1.5 they say that "...
- 10.7k
4
votes
1
answer
469
views
How to calculate $\mathrm{TP}(\mathbb{F}_p[t])$?
$\DeclareMathOperator\TP{TP}$I am trying to learn about topological periodic cyclic homology following the notes:
https://www.uni-muenster.de/IVV5WS/WebHop/user/nikolaus/Papers/Lectures.pdf
https://...
- 959
4
votes
2
answers
338
views
Attaching cells of different dimensions at once in a CW-complex
Let $X$ be a CW-complex and $X^m$ it's $m$-skeleton. I think that for any $n\geq 2$ and $1\leq r\leq n-1$ it should be possible to obtain $X^{n+r}$ directly from $X^n$ via a homotopy push-out
$$\...
- 15.2k
4
votes
2
answers
598
views
Attaching cells of different dimensions at once in a CW-complex II
This question is related to Attaching cells of different dimensions at once in a CW-complex There, I didn't manage to formalize the idea I had in mind, and ended up with a question whose answer was ...
- 15.2k
4
votes
0
answers
133
views
Equivalence of Flat Fiber Bundles vs Equivalence of Group Actions on the Fiber
Let's consider all flat fiber bundles with base space $B$ and fiber $F$, where $B$ and $F$ are compact and at least CW-complexes. (perhaps even topological/smooth manifolds if that helps)
All those ...
- 404
4
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1
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456
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Homotopy groups of K3
Let X be a K3 surface and $Y=X/\mathbb{Z}_2$, an Enrique surface.
Long exact sequence of homotopy groups corresponding to fiberaion $\pi:X\to Y$, says that $\pi_2(X)=\pi_2(Y)$, while we know $H_2(X)$ ...
4
votes
0
answers
74
views
Determinant twist and $Pin _{\pm}$ structure on $4k$-dimensional bundles [Reference request]
Consider the automorphism $\varphi$ of $O(2n)$ given by $\varphi (A)=det(A)\cdot A$. The induced map in cohomology $H^*(BO(2n))$ sends $w_2$ to $w_2+(n+1)w_1^2$ (a proof is given at the end of the ...
- 3,051
4
votes
2
answers
536
views
Explicit tetrahedralizations of closed 3-manifolds and connections between convex polytopes and hyperbolic knot complements
There are three consistent ways to glue opposite faces of a dodecahedron to get a closed 3-manifold. With a $\frac{1}{10}$ twist we receive the Poincaré dodecahedral space, with a $\frac{3}{10}$ twist ...
- 1,441
4
votes
1
answer
447
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Complex projective manifold with an antiholomorphic involution
Let $M$ be a complex projective manifold with an antiholomorphic involution. Can $M$ be defined by equations with real coefficients then?
user164740
4
votes
1
answer
183
views
When can a generalized connected sum be aspherical
Let $M$ and $N$ be compact $n$-dimensional manifolds with a common (nicely embedded) compact submanifold $S$ (we may assume that the inclusion of $S$ in $M$ and $N$ is $\pi_1$-injective). Let $M\#_S N$...
- 311
4
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1
answer
361
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Localization at the Johnson-Wilson spectrum and rationalization
Is there a clean proof that the $L_n$, localization at $E(n)$, is simply rationalization (i.e. $L_0$) on Eilenberg-MacLane spectra? Eric Peterson asked this here, but I haven't seen an answer.
user62675
4
votes
2
answers
847
views
Künneth formula for Bredon cohomology theory
Let $G$ be a finite group. Let $X$ and $Y$ be two $G$-CW complexes with known integer graded $G$-equivariant Bredon cohomology with constant coefficient systems. Is there any Künneth formula for this ...
- 745
4
votes
4
answers
826
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When is the quotient by an $n$-fold loop space an $m$-fold loop space?
Given a map of $n$-fold loop spaces $X\to Y$, we can take the homotopy cofiber, denote it $Y/X$ (all spaces here will also have a base point, and all maps pointed). I have some basic questions about ...
- 10.5k
4
votes
0
answers
945
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Simplicial chain complex with ordered simplices
Let $X$ be an abstract simplicial complex. Recall that the usual simplicial chain complex for $X$ is defined as follows. Let $C_k(X)$ be the quotient of the free abelian group on formal symbols $[...
- 41
4
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1
answer
321
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Isomorphism of coends
This is a follow-up to this question:
Reduction to graph subgroups for Bredon homology when the $G_1\times G_2$ is $G_2$-free
In his (very nice) answer Gregory Arone stated the following fact. Let $Q:\...
- 1,759
4
votes
1
answer
326
views
Rational homotopy type of rational mapping spaces
I was interested in the question of figuring out the rational homotopy type of mapping spaces (regular or rational) between two algebraic varieties over $\mathbb{C}$. I encountered the following paper ...
- 5,901
4
votes
0
answers
94
views
A dimension condition on the cohomology of a homogeneous space
The rational cohomology of a homogeneous space $G/K$ admits a homomorphism from $H^*(BK)$ induced from the classifying map $G/K \to BK$ of the principal $K$-bundle $G \to G/K$. Assume the Lie group is ...
- 2,995
4
votes
1
answer
339
views
Is there a class of simplicial sets whose weak homotopy type is preserved by symmetrization?
We have the categories, $S$, of simplicial sets and $SS$, of symmetric simplicial sets (whose simplices are unordered). There are functors:
$H:S\to SS$ forgetting the ordering on simplices and
$L:...
- 81
4
votes
3
answers
1k
views
Circle bundles over $CP^1$ and self-intersection number of $CP^1$ embeddings
If $X$ is a compact oriented surface in a 4-dimensional oriented manifold $M$, then the self-intersection number $X^2$ of $X$ is given by the integral over $X$ of the Euler class of the normal bundle. ...
- 639
4
votes
1
answer
522
views
Can every manifold be represented as a quotient
My question is "inspired" by the uniformization theorem for Riemmannian surfaces and this post.
Suppose that $X$ is connected (finite-dimensional) topological manifold without boundary. ...
- 5,405
4
votes
2
answers
848
views
Generalized Jordan theorem and winding number
By the generalized Jordan theorem any continuous injective map
$S^{n-1} \hookrightarrow R^n$ splits $R^n$ into two regions, one being bounded (interior) and the other one unbounded (exterior). It ...
- 1,875