All Questions
Tagged with gt.geometric-topology at.algebraic-topology
1,145 questions
19
votes
5
answers
2k
views
References for Eilenberg-Zilber shuffle product
Most of the treatments I can find in the literature for the Eilenberg-Zilber shuffle product approach it from the point of view of simplicial sets (including the original Eilenberg-MacLane paper). I ...
- 5,489
19
votes
1
answer
862
views
Diffeomorphism groups of h-cobordant manifolds
Do we have specific examples of h-cobordant smooth manifolds $M$ and $M'$ such that $\operatorname{BDiff}(M) \not \simeq \operatorname{BDiff}(M')$? Perhaps something can be said in terms of K-theory ...
- 5,839
19
votes
0
answers
649
views
Bernoulli & Betti numbers (of manifolds) and the prime 34511
The purpose of this question is to resolve a mystery surrounding the prime 34511 that has got me bogged down for a while now. If you only care about the number theory and not the motivation coming ...
- 11.9k
19
votes
0
answers
410
views
are there high-dimensional knots with non-trivial normal bundle?
Does there exist a smooth embedding $\varphi\colon S^k\to S^n$ such that $\varphi(S^k)$ has non-trivial normal bundle?
I looked at some of the old papers by Kervaire, Haefliger, Massey, Levine but I ...
- 2,797
18
votes
4
answers
1k
views
Explicit constructions of K(G,2)?
Recall that an Eilenberg-Maclane space $K(G, n)$ is characterized by $\pi_i(K(G,n)) = G$ if $i=n$ and is trivial otherwise. (Of course $G$ should be abelian if $n>1$.)
Let $G$ be a finite abelian ...
- 12.8k
18
votes
1
answer
1k
views
Simply connected finite CW-complex with only finitely many nontrivial homotopy and homology groups
Let $X$ be a simply connected finite CW-complex such that all but finitely many of its homotopy groups and its homology groups (with $\mathbb Z$ coefficients) are 0.
Is $X$ then necessarily ...
- 11.9k
18
votes
1
answer
1k
views
Diffeomorphisms vs homeomorphisms of 3-manifolds
For a smooth 3-manifold $M$, is the natural map from the space of diffeomorphisms of $M$ to the space of homeomorphisms of $M$,
$${\sf Diff}(M) \longrightarrow {\sf Top}(M),$$
a weak homotopy ...
- 967
18
votes
2
answers
1k
views
What is the generator of $\pi_9(S^2)$?
This is more or less the same question as
[ What is the generator of $\pi_9(S^3)$? ], except what I would like to know is if it is possible to describe this map in a way
not only topologists can ...
- 6,891
18
votes
1
answer
1k
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Wu formula for manifolds with boundary
The classical Wu formula claims that if $M$ is a smooth closed $n$-manifold with fundamental class $z\in H_n(M;\mathbb{Z}_2)$, then the total Stiefel-Whitney class $w(M)$ is equal to $Sq(v)$, where $v=...
- 1,329
18
votes
2
answers
1k
views
formula for Eta invariant
Hirzebruch's signature formula is not valid for manifolds with boundary.
An error term is introduced by Atiyah-Patodi-Singer to fix it.More precisely:
$$sign (M)=L(M)[M]+\eta(\partial M)$$
Yet ...
- 259
18
votes
2
answers
790
views
The kernel of the map from the handlebody group to Outer automorphisms of a free group
Let $K$ be a compact oriented 3-dimensional handlebody of genus $g$. The group $H_g$ of isotopy classes of diffeomorphisms of $K$ is called the handlebody group. (It embeds as a subgroup of the ...
- 6,229
18
votes
1
answer
521
views
How aggressive is the fibrant replacement of $\mathrm{Bord}_n$?
Lurie (On the Classification of Topological Field Theories), with some corrections by Calaque and Scheimbauer (A note on the $(\infty,n)$-category of cobordisms), famously constructed a symmetric ...
- 54.6k
18
votes
1
answer
872
views
Oriented cobordism classes represented by rational homology spheres
Any homology sphere is stably parallelizable, hence nullcobordant. However, rational homology spheres need not be nullcobordant, as the example of the Wu manifold shows, which generates $\text{torsion}...
- 11.9k
18
votes
1
answer
1k
views
Fundamental groups of the spaces of rational functions
Here is a question which I asked myself (and couldn't answer) while reading "The topology of spaces of rational functions" by G. Segal.
Let $X$ be a smooth complete complex curve (=a compact Riemann ...
- 23.5k
18
votes
1
answer
1k
views
Is the restriction map for embeddings of manifolds with boundary a fibration?
Let $M$ and $W$ be smooth manifolds (possibly with boundary) and $V\subseteq W$ a submanifold. We have a map between embedding spaces
$$Emb(W,M)\rightarrow Emb(V,M)$$ given by restriction.
Richard ...
- 2,974
18
votes
1
answer
1k
views
On the definition of A-theory
Waldhausen's A-theory is a version of algebraic K-theory of spaces. Concretely, for a (pointed) space $X$, he considers the 'Waldhausen category' $\mathcal R_f(X)$ of finite retractive CW-complexes ...
- 11.9k
18
votes
1
answer
943
views
Do chains and cochains know the same thing about the manifold?
This question was inspired by Poincaré quasi-isomorphism
Let $M$ be a closed oriented $n$-manifold. The cap product with the fundamental class of $M$ induces an isomorphism $H^i(M,\mathbf{Z})\to ...
- 23.5k
18
votes
1
answer
797
views
Do mapping classes have gonality?
(This question was discussed by people at the PCMI workshop on moduli spaces, without any clear resolution, so I thought I'd throw it open to MO.)
The hyperelliptic mapping class group is (by ...
- 19.2k
18
votes
0
answers
1k
views
What is the strongest nerve lemma?
The most basic nerve lemma can be found as Corollary 4G.3 in Hatcher's Algebraic Topology:
If $\mathcal U$ is an open cover of a paracompact space $X$ such that every nonempty intersection of ...
- 81
18
votes
0
answers
496
views
Orientation-reversing homotopy equivalence
If there is an orientation-reversing homotopy equivalence on a closed simply-connected orientable manifold is there an orientation-reversing homeomorphism?
It is not true, for instance, that if there ...
user164740
18
votes
0
answers
328
views
"High-concept" explanation for proof of a theorem of Ochanine?
See Akhil Mathew's notes on Ochanine's theorem for elliptic genera here and here.
Let $\phi: \Omega_{SO} \to \Lambda$ be a genus. We might ask when $\phi$ satisfies the following multiplicative ...
user82367
17
votes
3
answers
954
views
Can an oriented closed $n(\geq 2)$-dimensional manifold be smoothly embedded in $\mathbb{R}^{2n-1}$?
Can anyone provide me with an example of an orientable closed manifold $M$ of dimension $n\geq 2$, which cannot be smoothly embedded in $\mathbb R^{2n-1}$?
I know these cannot exist for $n=1$, i.e. $S^...
- 3,751
17
votes
2
answers
1k
views
Suspension of a topological space
Let $X$ be a topological space such that its suspension is a topological manifold. Can we prove that $X$ itself is a topological manifold?
- 2,535
17
votes
3
answers
1k
views
Homology generated by lifts of simple curves
Let $\Sigma$ be a compact connected oriented surface and $p:\tilde{\Sigma}\to\Sigma$ a finite regular cover.
Consider the set $\Gamma$ of simple closed curves on $\tilde{\Sigma}$ obtained as a ...
- 1,181
17
votes
2
answers
1k
views
Homotopy groups of Diff(X) and Homeo(X)
For a compact closed smooth manifold $X$, the group Diff(X) has a natural homomorphism $\Phi$ to the homeomorphism group Homeo(X). If $X$ has dimension at least $5$, I'm looking for some general ...
- 19.4k
17
votes
2
answers
2k
views
homotopy type of embeddings versus diffeomorphisms
Previously, I asked a question on mathoverflow comparing smooth embeddings and diffeomorphisms, which received a very interesting and somewhat unexpected answer by Agol. I now ask a further question ...
- 6,210
17
votes
3
answers
1k
views
Codimension zero immersions
Given an immersion of the n-1-sphere into a (closed) n-manifold, when does it extend to an immersion of the n-disk?
Remark: If the sphere had dimension k smaller than n-1, then such an immersion ...
- 10.4k
17
votes
1
answer
898
views
Does there exist a closed manifold with vanishing reduced rational cohomology but nonvanishing odd torsion cohomology?
Question: Let $p$ be an odd prime. Does there exist a closed manifold $M$ with $\widetilde H^\ast(M; \mathbb Q) = 0$ but $\widetilde H^\ast(M; \mathbb F_p) \neq 0$?
When $p = 2$, an example is given ...
- 63.9k
17
votes
2
answers
2k
views
Involutions of $S^2$
are there some complete results on the involutions of 2 sphere?
at least I have three involutions:
(let $\mathbb{Z}_2=\{1,g\}$,and $S^2=\{(x,y,z)\in\mathbb{R}^3|x^2+y^2+z^2=1\}$)
1.$g(x,y,z)=(-x,-y,-...
- 291
17
votes
1
answer
506
views
Topology of the space of embedded genus $g$ surfaces in $S^3$
Consider the space of smoothly embedded genus $g$ surfaces in the 3-sphere under the $C^\infty$ topology:
$$\mathcal E_g:=\operatorname{Emb}(\Sigma_g,S^3)/{\operatorname{Diff}(\Sigma_g)}$$
where $\...
- 525
17
votes
1
answer
683
views
Relationship between Smith's special homology groups and equivariant homology theory
EDIT: Tyler Lawson's answer was so nice that I was inspired to rewrite the notes discussed below to use Bredon homology in the definition of the Smith special homology groups. The original version is ...
- 44.8k
17
votes
1
answer
574
views
Simply connected slices
Assume $\Omega$ is an open set in $\mathbb R^3$
such that the intersection of $\Omega$ with any horizontal plane is simply connected.
Can you prove that $\Omega$ is simply connected?
(Note that ...
- 45k
17
votes
3
answers
1k
views
The second homotopy group of a simple CW-complex
Let $X$ be a CW-complex with
one 0-cell
two 1-cells
three 2-cells
no cells in dimensions 3 or higher.
Is it always true that $\pi_2(X)\ne 1$?
- 1,181
17
votes
1
answer
981
views
Smooth 4-manifolds with $E_8$ intersection form
Does there exist a closed orientable smooth 4-manifold $M$ whose intersection form is the $E_8$-form? Here by the intersection form I mean the $\mathbb{Z}$-valued bilinear form on $H^2(M;\mathbb{Z})/\...
- 193
16
votes
10
answers
3k
views
Orbifold fundamental group in terms of loops?
In chapter 13 in his notes on 3-manifolds, Thurston defines the orbifold fundamental group to be the group of deck transformations of the universal cover of the orbifold. He also makes a statement "...
- 13.6k
16
votes
10
answers
6k
views
Undergraduate Topology
I am developing an introductory topology course for undergraduates, and I am wondering what topics to cover. At my institution, real analysis is not a prerequisite for the course, so it is more than ...
16
votes
3
answers
3k
views
open problems in Seiberg-Witten Theory on 4-Manifolds
What are some of the open problems in Seiberg-Witten Theory on 4-Manifolds.I tried googling but couldn't any. I tried googling it, but couldn't find any resources.The places where I can a survey or ...
16
votes
1
answer
1k
views
A concrete realization of the nontrivial 2-sphere bundle over the 5-sphere?
Since $\pi_4 (PU(2)) = \pi_4 (SO(3)) = {\mathbb Z}_2$, the two-element group,
we know that half of the two-sphere bundles over the 5-sphere $S^5$ are trivial
and the other half are non-trivial and ...
- 8,336
16
votes
3
answers
940
views
Relationships between homology maps of cobordant manifolds
Let $W$ be a cobordism between $n$-manifolds $M$ and $N$, and $f\colon W \mapsto X$ be a map to some manifold $X$.
Does anybody know of any nice examples of general relationships between the images ...
- 161
16
votes
2
answers
820
views
Klee's trick --- more applications
In his "Some topological properties..." (1955), Klee gave a construction (simple and beautiful) of an isotopy $h_t\colon\mathbb{R}^{2\cdot n}\to \mathbb{R}^{2\cdot n}$ which moves any compact set $K$ ...
- 45k
16
votes
3
answers
1k
views
SO(3) action on (simply connected) 6 manifold with discrete fixed point
If a 6-dimensional orientable smooth manifold $M$ admits a smooth effective $SO(3)$ action with discrete fixed point set, what can we say about the topology of $M$? What if we assume that in addition ...
- 591
16
votes
2
answers
605
views
What is the weakest negative curvature condition ensuring a manifold is a $K(G,1)$?
The only statement I'm sure of is that any hyperbolic or Euclidean manifold is a $K(G,1)$ (i.e. its higher homotopy groups vanish), since its universal cover must be $\mathbb H^n$ or $\mathbb E^n$. ...
- 63.9k
16
votes
3
answers
2k
views
When does a CW-complex of dimension 2 embed in $\Bbb R^4$?
Let $X$ be a finite CW-complex of dimension two having just one 0-cell
(+ finitely many 1-cells + finitely many 2-cells).
Is it true that X can be embedded in $\Bbb R^4$?
If true, is it due to ...
16
votes
1
answer
797
views
Do vanishing characteristic classes of the tangent bundle imply a manifold is stably frameable?
Suppose we know that all stable characteristic classes of the tangent bundle of a manifold $M$ vanish, i.e. the map $f:M\rightarrow BO$ stably classifying the tangent bundle is trivial on cohomology, ...
- 323
16
votes
1
answer
505
views
How many cells needed to build the classifying space $BG$?
Let $G$ be a finitely presented group of cohomological dimension $n$.
Apart from the unresolved ambiguity pertaining to the Eilenberg--Ganea conjecture, it is known that we can find an $n$-dimensional ...
- 11.9k
16
votes
0
answers
425
views
Is the oriented bordism ring generated by homogeneous spaces?
I am trying to find a Riemannian geometrically well-understood set of generators of the oriented bordism ring, including the torsion parts. By a set of generators, I mean that the set generates the ...
- 587
16
votes
0
answers
325
views
Rational equivalence of smooth closed manifolds
All spaces below will be assumed simply connected. A continuous map is a rational equivalence if it induces an isomorphism of the rational homology groups. Two spaces are rationally equivalent if they ...
- 23.5k
16
votes
0
answers
1k
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Connected sum is well-defined for surfaces, proof?
EDIT: So my question is distinct from the question asked here because I am asking an easier question. Why should we have to invoke something as powerful as the "Annulus Theorem" to show that the ...
- 169
15
votes
5
answers
3k
views
Generalization of winding number to higher dimensions
Is there a natural geometric generalization of the winding number to higher dimensions?
I know it primarily as an important and useful index for closed, plane curves
(e.g., the Jordan Curve Theorem),
...
- 151k
15
votes
3
answers
1k
views
Linking topological spheres
Is there a simple proof of the fact that:
If $A\subset S^3$ is homeomorphic to $S^1$, then there is a circle $B$
embedded into $S^3\setminus A$ that such that the circles $A$ and $B$
are ...
- 28k