All Questions
1,240 questions
6
votes
3
answers
2k
views
Fake projective spaces
I'm looking for examples of "fake" projective spaces.
Question: Are there smooth manifolds other than $\mathbb{C}\mathbb{P}^n$ whose cohomology ring is the truncated polynomial ring $\mathbb{K}[h]/h^...
- 61
6
votes
2
answers
539
views
Morphisms every pushout of which is a weak equivalence
Let $M$ be a category equipped with a class of weak equivalences $W$. Is there a name for a morphism $f$ such that every pushout of $f$ (including, of course, $f$ itself) is a weak equivalence?
For ...
- 66.8k
6
votes
0
answers
189
views
Complete invariant of filtered chain complexes under chain homotopy equivalence
Betti numbers are a complete invariant of chain complexes of vector spaces modulo chain homotopy equivalence.
Can we similarly find complete invariants for (say, finite dimensional) filtered chain ...
- 2,772
6
votes
0
answers
244
views
Borel vs genuine equivariant cohomology in quantum field theory
A lot of important work in quantum field theory involves Borel equivariant cohomology of certain geometric objects, usually with the goal of computing integrals over some complicated moduli stack. In ...
- 1,969
6
votes
1
answer
1k
views
Is geometric realization of the total singular complex of a space homotopy equivalent to the space?
Let $X$ be a topological space and let $|Sing(X)|$ be the geometric realization of the total singular complex of $X$.
Then $|Sing(X)|$ is a CW complex with one cell for each non-degenerate singular ...
- 1,414
6
votes
0
answers
384
views
Mod 3 Moore spectrum
I only know through stories that mod 3 moore spectrum is not associative. I do not know of any proof. I have been informed that Toda had proved it in the paper "Extended $p^{th}$ power". I was not ...
- 2,023
6
votes
1
answer
246
views
Topology of hypersurface of sphere fixed by homeomorphic involution
I'm not an topologist, so I apologize in advance if this is a silly question.
I have the following situation, let $\mathbb{S}^n$ (for $n\geq 3$) be the standard (smooth if it matters) $n$-sphere and $...
- 1,149
6
votes
1
answer
1k
views
Transporting model structures via adjunctions
Hello,
If $F$ is a left adjoint between $C$ and $D$, and $D$ has a model structure; We can define cofibrations and equivalences in $C$ to be those that are so after applying $F$. What are criterions ...
- 5,562
6
votes
1
answer
515
views
Numerable covers from the point of view of Grothendieck topologies
Let $G$ be a topological group. Recall that its classifying space $BG$ is a CW-complex which is the base of a locally trivial principal bundle of group $G$, with contractible total space $EG$. It ...
- 12.9k
6
votes
3
answers
395
views
Decomposable maps of half-smash products
[Cross-posted from MSE]
For a pointed space $X$ and unpointed space $Y$, recall the half-smash product $X\rtimes Y=X\land Y_+=(X\times Y)/(\ast\times Y)$. For unpointed spaces $X,Y$ and a pointed ...
- 508
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
5
votes
2
answers
923
views
Does anyone know where I can get a copy of Gaunce Lewis's thesis?
Tracing through a trail of references I found myself needing something proven in Appendix A of "The Stable Category and Generalized Thom Spectra" by Gaunce Lewis (I believe this was his thesis at ...
- 30.3k
5
votes
1
answer
298
views
Interpolating between the flat and smooth affine lines in spectral algebraic geometry
Consider the following construction (which came up recently in a question about "spectral exterior algebras"):
Pick a ring spectrum $R$ and consider the $\infty$-category $\mathsf{Mod}_R$ ...
- 11.8k
5
votes
1
answer
312
views
What can be said about the map $K(n)_\ast(X) \to K(n)_\ast(\tau_{\leq n}X)$ when $X$ is a finite complex?
Ravenel and Wilson showed that $K(\mathbb Z / p^j,q)$ is $K(n)$-acyclic for any $q \geq n+1$, and that $K(\mathbb Z, q)$ is $K(n)$-acyclic for $q \geq n+2$. It follows that $K(A,q)$ is $K(n)$-acyclic ...
- 64k
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
2
answers
308
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Is this subset of matrices contractible inside the space of non-conformal matrices?
Set $\mathcal{F}:=\{ A \in \text{SL}_2(\mathbb{R}) \, | \, Ae_1 \in \operatorname{span}(e_1) \, \, \text{ and } \, \, A \, \text{ is not conformal} \,\}$, and
$\mathcal{NC}:=\{ A \in M_2(\mathbb{R}) \...
- 6,741
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
553
views
Homotopy problem for infinite dimensional topological space II
This post here is a specification of this post.
Let $(X_{n},d_{n})_{n \in \mathbb{N}}$ be a sequence of intrinsic metric spaces verifying :
$X_{n}$ have topological dimension $n$.
$X_{n+1}$ is n-...
5
votes
1
answer
957
views
To what extent is a vector bundle on a manifold with boundary determined by its restriction to the interior?
Let $M$ be a manifold with boundary $\partial M$ and interior $M_0$. Let $E\rightarrow M_0$ be a fixed vector bundle. How many extensions of $E$ to a vector bundle $E'\rightarrow M$ are there, up to ...
- 195
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
2
answers
509
views
Example of a space X exhibiting the Landweber non-exactness of the additive formal group over the integers?
Landweber exactness gives a criterion for when a complex oriented cohomology theory $E$ can be recovered from the formal group law over $E_{*}$ determined by the complex orientation. That is it gives ...
- 1,150
5
votes
2
answers
355
views
Do infinite products commute with trivial cofibrations, for simplicial sets?
I'm reading Voevodsky and Morel's book '$\mathbb{A}^1$-homotopy theory of schemes'. In Remark 3.1.15, it says that for any simplicial fibrant sheaf $F$ and open sets $U\subseteq V$, $F(V)\to F(U)$ is ...
- 918
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2
answers
219
views
Reedy model structures on oplax limits
Suppose $R$ is a category and $F:R\to Cat$ is a functor (or pseudofunctor). The oplax limit of $F$ is the category whose objects consist of an object $x_r \in F(r)$ for all $r$ together with a ...
- 66.8k
5
votes
2
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1k
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Is a Lie group equivariantly formal under conjugation by a maximal torus?
Given an action of a group $G$ on a topological space $X$, the associated homotopy quotient is $$X_G := (EG \times X)/G,$$ where $EG$ is the total space of a universal principal $G$-bundle $EG \to BG$ ...
- 2,995
5
votes
1
answer
337
views
Oddness of intersection form of surface bundle
Let $\Sigma_g$ be a Riemannian surface of genus $g$. Let $M^4$ be a surface bundle: $\Sigma_g \to M^4 \to \Sigma_h$. When $g=1$, $M^4$ is called a torus bundle.
My question: is there a torus bundle ...
- 4,826
5
votes
0
answers
242
views
characteristic classes of a covering space with symmetric group action
Let $S_n$ be the $n$-th symmetric group. Suppose we have a $n!$-sheeted covering space
$$
S_n\to M\to M/S_n
$$
where $M$ is a manifold. Let $\mathbb{K}$ be the real numbers $\mathbb{R}$, complex ...
- 2,223
5
votes
2
answers
657
views
Homotopy equivalences preserving structure
Suppose I have $X=X_1\cup X_2\cup…\cup X_n$ and $f:X \to Y$ where $Y$ has a similar decomposition.
Suppose I know that
$f | X_{i_1}\cap…\cap X_{i_r} \to Y_{i_1}\cap…\cap Y_{i_r}$ is a homotopy ...
- 51
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
4
answers
3k
views
Nontrivial examples of non-trivial principal circle bundles
It is a well known fact that (isomorphism classes of) principal $S^1$-bundles over a base space $B$ are classified by $B$'s second integral cohomology, $H^2(B;\mathbb{Z})$.
There is always the ...
- 583
5
votes
3
answers
1k
views
Mayer-Vietoris Sequence for Arbitrary Bicartesian Square of Spectra
Can anyone tell me if there is a Mayer-Vietoris sequence for an arbitrary homotopy pushout (hence homotopy pullback) of spectra and an arbitrary (co)homology theory. If this comes from some easy way ...
- 10.5k
5
votes
0
answers
140
views
Globalising fibrations by schedules
In a paper in Fund Math 130.2 (1988): 125-136. http://eudml.org/doc/211719 Dyer and Eilenberg give an account of the local-global theorem for fibrations by proving a "Schedule Theorem" that, given a ...
- 12.3k
5
votes
1
answer
333
views
Proof of homotopic essential simple close curves are isotopic
In the book by Benson Farb and Dan Margalit A primer on mapping class groups, Princeton Mathematical Series 49. Princeton, NJ: Princeton University Press (ISBN 978-0-691-14794-9/hbk; 978-1-400-83904-9/...
- 119
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
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0
answers
135
views
Specify the embedding of Lie groups (via the representation map) precisely as the embedding of two differentiable manifolds
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
5
votes
2
answers
2k
views
Simple connectedness via closed curves or simple closed curves?
I've recently read some papers and books involving simply connected domains in Euclidean space (dimension at least 2), where domain is an open connected set. The usual definition is a (connected) set ...
5
votes
1
answer
330
views
characteristic classes of tangent bundle of 2-nd unordered configuration space
Given a (real or almost complex) manifold $M$, Let the 2-nd unordered configuration space be the quotient space
$$
B(M,2)=(M\times M\setminus\ \Delta)/\ \mathbb{Z}_2
$$
where
$$
\Delta=\{(m,m)\mid m\...
- 2,223
5
votes
1
answer
322
views
Is there a model structure for S-modules such that cofibrant operad-algebras forget to cofibrant S-modules?
In 1997, Elmendorf, Kriz, Mandell, and May wrote a book Rings, Modules, and Algebras in Stable Homotopy Theory in which they introduced the category of $S$-modules as a model for the stable homotopy ...
- 30.3k
5
votes
1
answer
619
views
Is every finite subcomplex of a contractible simplicial complex contained in a finite contractible subcomplex?
The question is as in the title:
Is every finite subcomplex of a contractible simplicial complex $K$ contained in a finite contractible subcomplex of $K$? What if we are allowed to take ...
- 2,104
5
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0
answers
204
views
homotopy type of box topology.
Suppose that $X$ is weakly equivalent to a point. Let $I$ be a set. Does $\prod_{i\in I}X$ weakly equivalent to a point, where $\prod_{i\in I}X$ is equipped with box topology ?
- 1,613
5
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0
answers
161
views
Twisted spin-bordism invariant and a possible Postnikov square from $d=2$ to $d=5$
This is a follow up more advanced question following Twisted spin bordism invariants in 5 dimensions. We follow the definitions in the earlier post.
I had discussed my computation of
$$
\Omega_5^{...
- 10.5k
5
votes
2
answers
559
views
Examples of $\mathbb{E}_{k}$-semiring spaces
Semirings, also called rigs, are rings without negatives: their underlying additive monoids are not groups (in other words, while rings are monoids in $(\mathsf{Ab},\otimes_{\mathbb{Z}},\mathbb{Z})$, ...
- 11.8k
5
votes
1
answer
287
views
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
1
answer
530
views
Reference for t-structures on stable model categories
What kind of definitions of t-structures
on stable model categories have been investigated in the literature?
Of course, one can always define a t-structure on a stable model category as a t-...
- 37.8k
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
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
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
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
728
views
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
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,\...
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