All Questions
9,056 questions
17
votes
2
answers
2k
views
Are the homology and cohomology Serre spectral sequences dual to each other?
If we use homology and cohomology over a field $k$, if a space has homology and cohomology groups of finite type in each degree, then $H_\ast(X;k)$ is dual to $H^\ast(X;k)$ using the universal ...
- 3,726
8
votes
3
answers
782
views
How to tell whether a compact manifold can be realized as a nontrivial fiber bundle?
Let $M$ be a compact closed path-connected manifold.
There is the Postnikov tower as in
http://en.wikipedia.org/wiki/Postnikov_system
which tells us $M$ can be realized as an inverse limit of a ...
- 68.9k
12
votes
2
answers
2k
views
Example of a CW complex not homeomorphic to the realization of a simplicial set?
I've often heard that we can give examples of CW complexes that aren't homeomorphic to the realization of any simplicial set (although I haven't heard that there exist Kan complexes that aren't ...
- 7,237
3
votes
0
answers
446
views
When does the normal bundle of a submanifold of Euclidean space admit a flat connection?
Given a smooth submanifold of $R^n$, I was wondering if there is a reasonably simple criterion for deciding whether its normal bundle admits a flat connection. I am not ruling out monodromy in the ...
- 313
1
vote
3
answers
2k
views
when cup product is a zero homomorphism [closed]
How to see that the cup products vanish on suspensions?
- 12.5k
8
votes
1
answer
652
views
How do branched coverings of complex surfaces "fit" with branched coverings of curves?
Since I'm used to working with algebraic $\pi_1$'s, which don't work well with surfaces, I find myself lacking geometric intuition when I attempt to do these types of purely geometric arguments. I'm ...
- 66.3k
1
vote
3
answers
2k
views
Reference for intersection and linking in algebraic topology
I have a feeling that I have seen some kind of theory of linking and intersection that applies in spaces that are not manifolds. I've found two kinds of theories in the books I've checked:
1) ...
- 23.5k
0
votes
1
answer
1k
views
About universal coefficient theorem
Let $(X,A)$ be a finite CW-pair $m=p^r$ for some prime $p$. Unspecified coefficient is in $\mathbb{Z}$.
From the universal coefficient theorem, We know that
$H^1(A;\mathbb{Z}_m)=\textrm{Hom} (H_1(A),...
- 55.9k
4
votes
1
answer
805
views
Homology dimension of the mapping class group of a surface with boundary
There is a result on the dimension bound for ${M_{g,n}}/S_n$, (the moduli space for Riemann surfaces of genus $g$ with $n$ marked points) that is
$H_{i}({M_{g,n}}/S_n)=0$, for $i\ge 6g-7+2n$ except $(...
- 1,499
4
votes
0
answers
732
views
Spectral sequence for reduced homology
In the Serre spectral sequence, is it true that we can replace homology by reduced homology? That is:
If $f:X\rightarrow B$ is a Serre fibration,with $F$ the fiber, then if
$\tilde E^2_{pq}=\tilde H_p(...
- 12.6k
15
votes
1
answer
496
views
Geometric models for classifying spaces of $GLn(Fq)$.
The title pretty much says it. In a follow-up to my question about alternating groups, does anyone know of a "geometric" model for $BGL_n(F_q)$? By "geometric" I mean "a space you would have heard ...
CommunityBot
- 1
5
votes
0
answers
517
views
A smooth twisted tensor product of dg algebras?
I want to consider a Z/2Z dg algebra. As an algebra, it is generated over $\mathbb{Q}$ by two elements where x is even and e is odd with the relations $xe=ex$ and $e^2=1$(this makes it in particular ...
- 3,758
8
votes
3
answers
1k
views
What is a principal refinement of a Postnikov system?
I've been reading the book of Hilton, Mislin, and Roitberg on Localization of Nilpotent Groups and Spaces. In Section II.2 they define a principal refinement at stage $n$ of a Postnikov system $$\...
- 55.9k
11
votes
1
answer
1k
views
Cubical cohomology and de Rham cohomology
Qiaochu's question on a discrete analogue of harmonic function theory reminded me of some thoughts I had a long time ago about the relationship between cubical cohomology and de Rham cohomology.
The ...
CommunityBot
- 1
11
votes
1
answer
964
views
What does this naive attempt at $S^1$-equivariant homology describe?
After reading Cohen and Voronov's notes on string topology, one can find the following construction: Suppose we have a topological space $X$ with continuous action of $S^1$. This means we have a map $\...
- 55.9k
3
votes
3
answers
439
views
Sequential colim vs sequential hocolim
Suppose we have some homotopical setting in which we can speak of homotopy colimits. The setting I have in mind at the moment is that of a compactly generated triangulated category with a model, but ...
- 55.9k
12
votes
1
answer
535
views
4-manifolds in the 4-sphere such that it, *and* its complement have unsolvable word problem
In an earlier thread I had asked whether or not one can find a smooth 4-dimensional submanifold of $S^4$ whose fundamental group has an unsolvable word problem. The answer is yes, and the reference ...
CommunityBot
- 1
3
votes
1
answer
265
views
Equivariant Surgery problem
I have a question about surgery.
Let $G= \mathbb{Z}_m \times \mathbb{Z}$ and $M$ be a oriented 3-manifold with G-action. i.e. There exists a map $f\colon M/G \to BG$, where $BG$ is classifying space.(...
- 339
8
votes
1
answer
1k
views
Do homotopy colimits always commute with homotopy colimits?
Do homotopy colimits commute with homotopy colimits? The setting I am thinking of is that of a triangulated category with a model, but it would be interesting to have more general answers as well. A ...
- 23k
9
votes
3
answers
706
views
references / general idea of kervaire invariant problem
There's a workshop at MSRI in a couple months on the Kervaire invariant problem that I'd really like to attend. I saw Hopkins speak about it a while back without understanding much of the talk, but I'...
- 3,726
9
votes
0
answers
1k
views
Weight filtration over the integers
This is a follow up question to Weight filtration for smooth analytic manifolds
As mentioned in that question, the integral cohomology of some smooth complex analytic manifolds is equipped with a ...
CommunityBot
- 1
38
votes
2
answers
13k
views
Explanation for the Thom-Pontryagin construction (and its generalisations)
In 1950, Pontryagin showed that the n-th framed cobordism group of smooth manifolds was equal to n-th stable homotopy group of spheres:
$$ \lim_{k \to \infty} \pi_{n+k}(S^k) \cong \Omega_n^{\text{...
CommunityBot
- 1
7
votes
3
answers
750
views
Jordan Curve Homotopy
Does there exist a notion of Jordan curve homotopy?
In particular, suppose we have two Jordan curves $C_0 : S^1 \rightarrow \mathbb{R}^2$ and $C_1 : S^1 \rightarrow \mathbb{R}^2$. When does there ...
- 55.9k
10
votes
2
answers
897
views
Applications of classifying thick subcategories
So, relatively recently, Balmer introduced this notion of a spectrum for a tensor triangulated category and used it to prove a generalization of a classification theorem done in several areas of ...
- 13.5k
8
votes
1
answer
730
views
Status of Hilbert-Smith conjecture and H-S conjecture for Hölder actions
The Hilbert-Smith conjecture states that
If $G$ is a locally compact group which acts effectively on a connected manifold as a
topological transformation group then is $G$ a Lie group.
It was ...
- 12.6k
2
votes
0
answers
512
views
Lifting criteria of covering space by using homology condition
Let $\pi\colon\tilde{X}\to X$ be a p-fold (regular) cyclic covering(p:prime) and $\mathcal{A} = \mathrm{Im}(\pi_* )$, where $\pi_* \colon H_1(\tilde{X};\mathbb{Z}_p) \to H_1(X;\mathbb{Z}_p)$ is ...
- 7,442
6
votes
1
answer
850
views
Computation of homology groups of $M_{g,n}$
First some definitions: $\bar{M_{g,n}}$ is Deligne-Mumford space, i.e., the moduli space of stable nodal complex projective curves of genus $g$ with $n$ marked points. It is a complex orbifold, $\...
- 6,229
27
votes
2
answers
4k
views
What's the current state of the classification of not-fully-extended TQFTs?
Recall that a $(k,k+1,\dots,k+n)$-TQFT is (supposed to be) a functor from the $n$-category whose $j$-morphisms are (isomorphism classes of) compact $(k+j)$-dimensional manifolds with boundary to some ...
- 17.8k
18
votes
3
answers
1k
views
Periodicity theorems in (generalized) cohomology theories
It is well-known that topological K-theory is blessed with the Bott periodicity theorem, which specifies an isomorphism between $K^2(X)$ and $K^0(X)$ (where $K^n$ is defined from $K^0$ by taking ...
- 13.5k
7
votes
1
answer
633
views
Intersection product in a manifold, taking values in one factor
In a joint paper that I am working on, we are interested in taking the intersection product $[X] \cap [Y]$ of the fundamental classes of two compact, oriented pseudomanifolds $X$ and $Y$ in a compact, ...
- 55.9k
13
votes
1
answer
1k
views
Representing cohomology of a sheaf à la Eilenberg-Maclane
Suppose that we are given a nice space $X$ and a sheaf of abelian groups $F$ on $X$. Fix an integer $n$. Then We have a contravariant functor from nice spaces over $X$ to abelian groups; Namely, to a ...
- 47.3k
4
votes
1
answer
314
views
Homology of a complex projective conic
Let $Q$ be a smooth conic (the zero set of a homogeneous degree 2 polynomial)
in $\mathbb{P}^2(\mathbb{C})$ and let $j:Q\rightarrow\mathbb{P}^2(\mathbb{C})$ be the
immersion in the projective plane. ...
- 12.6k
8
votes
1
answer
906
views
Extraordinary cohomology as a derived functor?
The purpose of this question is to find out whether one can view the Atiyah-Hirzebruch spectral sequence as a particular case of the "composition of derived functors" spectral sequence.
The Leray ...
- 2,145
3
votes
3
answers
4k
views
Compactness and Covering Spaces
Let p : Y -> X be an n-sheeted covering map, where X and Y are topological spaces. If X is compact, prove that Y is compact.
I realize that this seems like a very simple problem, but I want to stress ...
- 47.5k
9
votes
1
answer
1k
views
Are there homotopy equivalences that are not weak homotopy equivalences?
I can imagine a map $f: X\to Y$ which is a homotopy equivalence of unpointed spaces, but which is not a homotopy equivalence of pointed spaces, no matter what basepoint is chosen. That being the case,...
- 10.9k
19
votes
0
answers
773
views
Folk Functorial Figuring
In the CRM Proceedings & Lecture Notes Volume 50 "A Celebration of the Mathematical Legacy of Raoul Bott" Herbert Shulman writes (p. 48):
"[Bott] taught many of us to think functorially, like ...
- 1,126
2
votes
0
answers
486
views
Casson Gordon paper - Cobordism of classical knots
It is given in Progress in mathematics 62, Guillou and Marin book. In the proof of Lemma 4, They choose $\alpha$ and $r\in \mathbb{N}$ such that $h^r_*\colon H_1(X;Z_p)\to H_1(X;Z_p)$ satisfies $h^r_*(...
- 4,990
32
votes
1
answer
1k
views
"Affine communication" for topological manifolds
There is a situation that comes up regularly in algebraic topology when giving proofs of facts about manifolds, like Poincare duality and the like. The typical sequence goes like this:
Prove ...
CommunityBot
- 1
0
votes
2
answers
1k
views
Question about the fundamental group and homotopy equivalence
Let T be a two-dimensional torus and Y be the one point
compactification of a two dimensional sphere ($S^2$) minus three points.
I have to prove:
1)they have the same fundamental group
2)they are ...
- 12.6k
5
votes
1
answer
938
views
Verifying a technical lemma regarding homotopy pushouts in the theory of simplicial model categories
Important Edit: I e-mailed Jacob Lurie, and he said that the statement of condition (*) is incorrect as printed.
Here is the correct statement of (*):
For any cofibration $f:A\to B$ and any trivial ...
CommunityBot
- 1
4
votes
0
answers
1k
views
Associative binary operations on natural numbers
Which are all the associative binary operations on natural numbers ?
Certain results in this regard can be found in arxiv:math/0508215.
It appears that such associative operations cannot grow too fast....
- 118k
6
votes
1
answer
700
views
Fibrations of Simplicial sets
Hello,
Maybe it is too vague a question, but I would like to ask if anybody could say some explanatory words about the importance (for infinity category study) of studying all the kinds of fibrations ...
- 15.5k
16
votes
2
answers
2k
views
Torsion in K-theory versus torsion in cohomology
Inspired by this question, I wonder if anyone can provide an example of a finite CW complex X for which the order of the torsion subgroup of $H^{even} (X; \mathbb{Z}) = \bigoplus_{k=0}^\infty H^{2k} ...
CommunityBot
- 1
5
votes
1
answer
977
views
How can we show the spaces $M_{g}(n)$ and $M_{g, n}$ are homotopy equivalent?
How can we prove that the moduli space,$M_{g}(n)$, of genus $g$ Riemann surface with $n$ boundary components is homotopy equivalent to $M_{g,n}$, that is ,the moduli space of genus $g$ Riemann surface ...
- 1,499
9
votes
2
answers
4k
views
How does singular homology H_n capture the number of n-dimensional "holes" in a space?
This is a foundational doubt I have. How does singular homology H_n capture the number of n-dimensional holes in a space?
We disregard the case of $H_0$ as it has the very satisfactory explanation ...
- 2,992
11
votes
0
answers
561
views
How to get a Dehn-twist presentation of a periodic map of a Riemann surface?
Let $f:S\to S$ be a given periodic map of order $n$, and $(m_i,\lambda_i, \sigma_i)$ be the valency of the multiple point $p_i$ of $f$ ( $i=1,\cdots,r$ ).
A classical result says such $f$ is ...
- 8,246
8
votes
2
answers
1k
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visualizing what's going on in based homotopy theory, et al.
I'm reading J.P. May's Concise Course in Algebraic Topology, and I'm having a lot of trouble visualizing how things work in Chapter 8, "Based cofiber and fiber sequences". Of course this is pretty ...
- 2,992
2
votes
1
answer
412
views
How can I prove that the derived couple of the homotopy exact couple is an invariant?
I'm working on (yet) an(other) exercise from Mosher & Tangora's "Cohomology Operations and Applications to Homotopy Theory". This one is about the homotopy exact couple, which is defined for a ...
- 12.5k
3
votes
0
answers
341
views
Descent of singular cohomology
When proving that singular cohomology of an appropriate space $X$ equals sheaf cohomology of $X$ with "values" (does one say that?) in the sheaf $\mathbb{Z}_X$ of locally constant functions, the ...
- 319
2
votes
1
answer
847
views
Proof Sketch: The pullback of the inclusion of the 0th vertex into the standard n-simplex by a right fibration is a deformation retract (450 point bounty if answered by 2am EST)
I was not sufficiently clear on my last attempt at asking a similar (but not identical) question. Tom Goodwillie mentioned (in the accepted answer) that the question can be reduced to this one and ...
- 19.6k