All Questions
Tagged with gt.geometric-topology at.algebraic-topology
1,145 questions
13
votes
2
answers
900
views
References for Stiefel-Whitney class of Stiefel manifolds and Grassmannians
Let $M$ be a manifold. The total Stiefel-Whitney class of $M$ is defined to be the Stiefel-Whitney class of the tangent bundle $TM$
$$
w(M)=1+w_1(TM)+w_2(TM)+\cdots
$$
I want to find references for
$$
...
- 2,223
13
votes
2
answers
993
views
When is a classifying space a topological manifold?
Let $G$ be a discrete group and $BG$ some model for the classifying space of $G$. So $BG$ is an aspherical path-conected topological space.
Under which conditions is $BG$ a topological manifold or ...
- 471
13
votes
2
answers
791
views
"C choose k" where C is topological space
One day I read a generating function in a paper. For any "sufficietly nice topological space", $C$:
$$ \sum_{l \geq 0 } q^{2l}\chi(\mathrm{Sym}^l[C]) = (1 - q^2)^{-\chi(C)} = \sum_{l \geq 0} \binom{...
- 22.8k
13
votes
3
answers
966
views
Rational homotopy theory of a punctured manifold
Let $M$ be a smooth simply connected manifold and let $N$ be $M$ minus a point. Is it possible to construct an explicit Sullivan model for $N$ (i.e. a commutative differential graded algebra (cdga) ...
- 23.5k
13
votes
1
answer
518
views
Impossibility of realizing codimension 1 homology classes by embedded non-orientable hypersurfaces
Suppose we have an $n+1$-dimensional compact closed oriented manifold $M$ and an $n$-dimensional integral homology class $[\Sigma]\in H_n(M,\mathbb{Z})$ on $M.$ Then is it true that $[\Sigma]$ mod $2$ ...
- 587
13
votes
1
answer
914
views
Relation between moduli spaces and classifying spaces
I hope this question is suitable to be posted here on MO.
I wonder if there is a systematic relation between the notation of a classifying space, and the notion of a moduli space. I don't consider ...
- 3,173
13
votes
2
answers
904
views
Discrete Morse function from smooth one
Suppose I have a smooth manifold $M$ and a smooth Morse function $f$ on $M$. Is there a standard way to replace $M$ with a cell complex and $f$ with a discrete Morse function such the resulting ...
- 609
13
votes
1
answer
669
views
Spin TQFT's in dimensions (1+1)
I don't seem to be able to find anything written about Spin TQFT's in dimension (1+1). Does anyone know any references? Or is there some reason it is uninteresting?
- 18.2k
13
votes
1
answer
386
views
Realizing integral homology classes on non-orientable manifolds by embedded orientable submanifolds
Let $M^m$ denote a compact, non-orientable smooth manifold and $\nu$ an integral homology class of dimension $n$. I am interested in understanding the representability of $\nu$ by embedded, orientable ...
- 587
13
votes
1
answer
492
views
Chromatic orientability of manifolds
If a compact manifold $M$ with empty boundary is oriented with respect to all the connective Morava $K$-theories $k(n)_*$, localized at a prime $p$, can one conclude that $M$ is orientable with ...
- 11.1k
13
votes
1
answer
762
views
Smooth structures on $M\times S^1$
Let $M$ be a smooth $n$-manifold. Here are three constructions which produce manifolds which are homeomorphic to $M\times S^1$, but might not be diffeomorphic to it:
Take $M'\times S^1$, where $M'$ ...
- 858
13
votes
1
answer
459
views
Compact closed aspherical manifolds with vanishing second homology in all the covering spaces
I wonder if there exists a compact closed smooth aspherical manifold $M$ of dimension at least $4$, so that for any covering space $\tilde{M}$ over $M,$ we always have $H_2(\tilde{M},\mathbb{Z})=0$ ...
- 587
13
votes
1
answer
289
views
Powers of the Euler class, torsion free subgroup of Homeo($S^1$)
For any subgroup $G$ of $\text{Homeo}(S^1)$, we have the Euler class $\chi$ in the group cohomology $H^2(G;\mathbb{Z})$. One can think of this class as the pullback of the generator of $H^2(\mathrm{B}\...
- 1,003
13
votes
0
answers
223
views
Examples of manifolds with first nontrivial SW-class in degree 16 or bigger
As a module over the Steenrod algebra, $H^{\ast}(BO;\mathbb F_2) = \mathbb F_2[w_1, w_2, w_3, \dots]$ is generated by $w_{2^t}, t \geq 0$. Thus, the first nontrivial SW-class of any vector bundle $\xi ...
- 11.9k
13
votes
0
answers
319
views
Exotic smooth structures on Fano manifolds
If two Fano projective manifolds are homeomorphic are they diffeomorphic?
There are examples with one manifold being Fano and the other of general type (Barlow surfaces). Moreover, the number of ...
user164740
13
votes
0
answers
330
views
One periodic cohomology theories?
Is there a classification of one periodic cohomology theories? In other words, spaces $X$ such that $\Omega X \simeq X$? For example, $X=*$ and $X= K(G,0) \times K(G,1) \times \dots$ are examples. ...
- 5,839
13
votes
0
answers
290
views
A geometric interpretation of the odd-primary Kervaire elements
Let $\Omega^\mathrm{fr}_\ast \cong \pi_\ast S$ denote the graded ring of cobordism classes of framed manifolds, which, by the Pontryagin-Thom construction, is isomorphic (as a graded ring) to the ...
- 5,760
13
votes
0
answers
371
views
What is the cup-product structure like on a hyperbolic 5-manifold?
Let $X$ be a hyperbolic 5-manifold. Can there be any class in $H^2(X;\mathbf{Z})$ that is torsion and whose square in $H^4(X;\mathbf{Z})$ is not zero?
For example, are there hyperbolic 5-manifolds ...
- 4,949
12
votes
3
answers
2k
views
Determining homotopy classes [T^2, RP^2]
So I've been interested in computing homotopy classes of maps $T^2=S^1\times S^1$ to $\mathbb{R}\mathbb{P}^2$. So first, we can decompose $T^2$ into a cell complex with one zero cell, $S^1\vee S^1$ ...
- 757
12
votes
4
answers
832
views
$S^n \to S^m \to B$ bundle: possible?
Sphere bundles and bundles over spheres are everywhere and are excellent things to get one's hands dirty with.
(1a) But when can we have a bundle $S^n \to S^m \to B?$ It seems like requiring the ...
- 2,734
12
votes
2
answers
767
views
Unique almost complex structure up to diffeomorphism
For which closed smooth manifolds does the action of the diffeomorphism group on the set of almost complex structures have exactly one orbit?
For example it is true for $S^2$.
user164740
12
votes
3
answers
849
views
$A_{\infty}$-structure on closed manifold
Is there an exmaple of a closed smooth connected manifold $M$ having a structure of $A_{\infty}$-space (with unit) but $M$ is not homeomorphic to a compact connectd Lie group as space ?
Edit: First, ...
- 1,974
12
votes
1
answer
1k
views
Are there non-compact, non-smoothable manifolds?
There do exist manifolds which do not admit any smooth structure at all. But the only examples I've heard of are all compact.
Are there any non-compact, non-smoothable manifolds?
- 2,998
12
votes
2
answers
660
views
Vector bundle for prescribed Stiefel-Whitney classes
I hope this is not trivial.
Let $B$ be a nice topological space (paracompact, CW-complex or whatever you think is nice)
For $i=1,\ldots,n$ let $x_i \in H^i(B,\mathbb{Z}_2)$ be certain cohomology ...
- 2,554
12
votes
2
answers
1k
views
Stable homotopy groups of $RP^{\infty}$
Are the stable homotopy groups $\pi^s_i(\mathbb R P^{\infty})$ known for small $i$? In particular, I would be interested in the values for $i = 5,6$. A quick Internet search did not lead to anything.
- 11.9k
12
votes
3
answers
4k
views
How are fiber bundles, transition functions and principal bundles related?
Please read the edit below.
Is my understanding of this correct? Fix a sufficiently nice and connected topological space $B$ and a topological group $G$. A principal bundle $E\to B$ with structure ...
- 1,085
12
votes
3
answers
1k
views
Fixed point set of smooth circle action
Suppose $M$ is a connected closed smooth $d$-dimensional manifold, and suppose $S^1 = SO(2)$ acts smoothly on $M$. Then the fixed point set $Y = M^{S^1}$ will be a submanifold of $M$ of even ...
- 11.9k
12
votes
2
answers
1k
views
A Compact Manifold with odd Euler characteristic whose tangent bundle admits a field of lines
I understand that the top Stiefel Whitney class is an obstruction for the tangent bundle of a manifold to have a trivial line sub-bundle. I am looking for a counterexample when removing the word "...
- 953
12
votes
2
answers
2k
views
The Alexander polynomial of a slice knot, Reidemeister tosion, Whitehead group
My question is about the Alexander polynomial of a slice knot.
For a slice knot $K$,
Fox-Milnor and Terasaka proved that
$$ \Delta_{K}(t) \doteq f(t) f(t^{-1})$$
for some polynomial $f(t) \in \...
- 391
12
votes
1
answer
566
views
Are $K(\pi_1,1)$ tangentially homotopy equivalent?
Is it known whether any two smooth, compact manifolds $X \simeq K(\pi_1,1) \simeq Y$ are tangentially homotopy equivalent, i.e. the pullback of the tangent bundle of $Y$ along some smooth homotopy ...
- 5,839
12
votes
3
answers
872
views
Can triangulations (or some related combinatorial structure) distinguish smooth structures on $RP^4$?
There are exotic versions of $RP^4$, constructed by Cappell-Shaneson, which are homeomorphic but not diffeomorphic to the standard $RP^4$. One way to distinguish them is via the $\eta$ invariant of $...
- 545
12
votes
3
answers
757
views
Quotient of the hyperbolic plane with respect to commutator group of $\pi_1(\Sigma_g)$
Let $\Sigma_g$ be a Riemann surface of genus $g\geq 2$ and $G=\pi_1(\Sigma_g)$.
Let $\pi\colon \mathbb{H}\to \Sigma_g$ be the universal covering map. What kind of surface is $\mathbb{H}/[G,G]$?
...
12
votes
2
answers
674
views
Cohomology of representation varieties
Perhaps this question is too general then I am sorry about this.
My question is the following.
Let $\pi$ be the fundamental group of a compact surface of genus $g$ (with if necessary $n$ punctures) ...
- 121
12
votes
1
answer
2k
views
Spin structures on $S^1$ and Spin cobordism
I'm trying to understand the 2 spin structures on the circle. Since the frame bundle for the circle is just the circle itself, Spin structures on $S^1$ correspond to double covers of $S^1$. There are ...
- 1,010
12
votes
1
answer
954
views
Does a self map from the wedge sum of two spheres have either a fixed point or a point of period 2?
Let $X$ be the wedge sum of two $2$-dimensional spheres and $f$ a continuous function from $X$ into $X$. Does $f$ have either a fixed point or a periodic point of order 2?
Thanks
- 121
12
votes
1
answer
4k
views
Cup products of connected sum
Hej,
I am interested in the cohomology ring of the connected sum $M \# N$ of two oriented manifolds $M$ and $N$ in terms of the corresponding cohomology rings of $M$ and $N$.
Mayer-Vietoris shows ...
- 121
12
votes
1
answer
840
views
Reference request: Topology on the space of smooth compact submanifolds
In Allen Hatchers short exposition of the Madsen-Weiss Theorem he defines the topology on the space $\mathcal{C}^n$ of smooth oriented properly embedded $d$-dimensional submanifolds of $\mathbb{R}^n$ ...
- 121
12
votes
1
answer
651
views
Does a triangulation without fixed simplex property always exist?
Suppose we are given a triangulable topological space $X$. If $X$ has the fixed point property (FPP), then obviously for every triangulation $K$ of $X$ and every simplicial map $f:K\to K$ a simplex $\...
- 2,104
12
votes
1
answer
489
views
Homological stability and Waldhausen A-theory
$\DeclareMathOperator{\Diff}{Diff}$
From the work of Galatius - Randall-Williams and Berglund - Madsen we have homological stability (with respect to g) of $B\Diff_\partial (W_{g,1})$ and rational ...
- 5,839
12
votes
3
answers
1k
views
A version of Lusternik–Schnirelmann category for good open covers
Recall that the Lusternik–Schnirelmann category (or LS-category) of a space is the integer $n$ such that there is an open cover by $n+1$ open sets which have nullhomotopic inclusions, and no such ...
- 35.5k
12
votes
1
answer
825
views
Stiefel-Whitney class of fibre bundles
Let $F,E,B$ be manifolds (we may assume them to be compact, without boundary if necessary) and $$F\to E\to B$$ be a fibre bundle. Suppose $$w(F), w(B),$$ the Stiefel-Whitney class of $F$ and $B$, are ...
- 2,223
12
votes
1
answer
2k
views
Conventions for definitions of the cap product
In singular (co)homology, if $\alpha\in C^*(X)$ and $x\in C_*(X)$, then the cap product $\alpha \cap x$ is generally defined by the following process:
Apply to $x$ the diagonal map $C_*(X)\to C_*(X\...
- 5,489
12
votes
1
answer
309
views
Dualizing module for $\operatorname{Aut}(F_n)$
In The complex of free factors of a free group (pdf at Hatcher's page), Hatcher and Vogtmann defined a simplicial complex $FC_n$ called the ``complex of free factors'' of the free group $F_n$. They ...
- 965
12
votes
1
answer
1k
views
classification of smooth involutions of torus
Let $\mathbb{Z}_2=\{1,g\},T^2=\{(e^{i\theta_1},e^{i\theta_2})\}$ and place $T^2$ in $\mathbb{R}^3$ as the locus of the rotation of $2\pi$ rads of the circle$\{(y,z)|(y-2)^2+z^2=1\}$ around $z$ axis.
...
- 291
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 ...
- 44.4k
12
votes
1
answer
372
views
Novikov-Wall non-additivity theorem with twisted coefficients
Let $Y$ be a compact manifold and let $\pi_1(Y) \to \mathbb{Z}^n= \langle t_1,\ldots,t_n\rangle$ be a homomorphism. Extend it to the group rings $\mathbb{Z}[\pi_1(Y)] \to \mathbb{Z}[ t_1,\ldots,t_n]$ ...
- 121
12
votes
1
answer
217
views
A variant of $\ell^2$-cochains
Suppose $X$ is an infinite countable CW complex which satisfies the following property: for all $k$-cells $e$, the number of $(k+1)$-cells incident to $e$ is at most $c_k$, where the latter is some ...
- 18.8k
12
votes
1
answer
1k
views
Stable normal bundle of a manifold
Hi,
in bordism-theory and many bordering areas one has the following construction: Given a manifold M (say closed for the purposes of this discussion and k-dimensional), we embed it into some $\...
- 333
12
votes
1
answer
818
views
The mathematics of tavern puzzles
I remember seeing a paper on the arxiv this year (which I cannot now find Edit: This paper: http://arxiv.org/abs/1208.6545, found by j.c.) proposing to study the linkage of rigid bodies such as tavern ...
- 3,359
12
votes
1
answer
725
views
Injectivity of the Dehn-Nielsen-Baer map?
If $S$ is a closed hyperbolic surface, is there an easy proof of the injectivity of the Dehn-Nielsen-Baer map from $\mathrm{Mod}(S)$ to $\mathrm{Out}(\pi_1(S))$, taking an element of the mapping class ...
- 10.1k