All Questions
Tagged with gt.geometric-topology at.algebraic-topology
1,145 questions
15
votes
3
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Unstable manifolds of a Morse function give a CW complex
A coauthor of mine and I want to use the following innocent looking statement in a forthcoming paper:
Statement. Let $M^{2n}$ be a compact manifold and let $f$ be a Morse function with critical ...
- 28.9k
15
votes
2
answers
2k
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The space of framed functions
Framed functions arose in the work of K. Igusa defining cohomology invariants for smooth manifold bundles (Igusa-Klein torsion). In the late 80's, he proved a strong connectivity result about the "...
- 2,734
15
votes
1
answer
755
views
Do $\mathbb{R}^n$ bundles have unit sphere bundles?
Recall there are multiple ways to define the unit sphere bundle of a vector bundle. One is by constructing a fiberwise vector space metric and declaring the sphere bundle to have fibers the unit ...
- 5,839
15
votes
2
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968
views
Semidirect product decomposition of the Borromean rings group
Let $X=S^3\setminus B$ be the link complement of the Borromean rings.
(source)
Then $G=\pi_1(X)$ has a presentation of the form
$$
G = \langle \; a,b,c \mid [a,[b^{-1},c]],\; ...
- 35.9k
15
votes
3
answers
1k
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Space of embeddings of circle in a surface
Let $S$ be a compact oriented surface of genus at least $2$ (possibly with boundary). Let $X$ be a connected component of the space of embeddings of $S^1$ into $S$.
Question : what is the ...
- 151
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1
answer
718
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When can a contractible 2-complex be embedded in R^3?
Let $X$ be a contractible 2-dimensional simplicial complex. Are there nice necessary and sufficient conditions for $X$ to be embeddable in $\mathbb R^3$? Clearly it is necessary that the link of ...
- 18.7k
15
votes
1
answer
541
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Where is the Steenrod Realization problem at?
I'm wondering if there is a more modern reference out there for the Steenrod Realization Problem than the book of Connor and Floyd?
Realizing homology classes in a manifold via embedded submanifolds, ...
- 44.4k
15
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1
answer
954
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Extending diffeomorphisms
Suppose we have a diffeomorphism $f:{\mathbb{S}}^n_{+}\to\mathbb{S}^n$ of class $C^1$ of the closed upper hemisphere onto a submanifold of $\mathbb{S}^n$ with boundary.
Question. Is it possible to ...
- 28k
15
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2
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973
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Infinity de Rham quasi-isomorphism
This question is similar to Do chains and cochains know the same thing about the manifold? in the sence that both deal with a natural "comparison" quasi-isomorphism that does not preserve the ring ...
- 23.5k
15
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3
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1k
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What do absolute neighborhood retracts look like?
In the course of filling in my map of non-pathological topology, I'd like to understand the class of ANRs (Absolute Neighborhood Retracts) as a sort of "neighborhood" of the class of CW complexes. ...
- 63.9k
15
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2
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927
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$\pi_4$ of simply-connected 4-manifold
In Baues "The homotopy category of simply conected 4-manifolds" there is some
algebraic description of $\pi_4(M^4)$ where $M^4$ is simply-connected closed 4-manifold, but this description is pure ...
- 5,053
15
votes
1
answer
2k
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Pseudomanifolds and Poincaré duality
1) A $n$-dimensional homology manifold is a topological space $X$ such that for any $x\in X$, the homology groups
$$H_p(X,X-x,\mathbb{Z})$$
are trivial unless $p=n$ where
$$H_n(X,X-x,\mathbb{Z})\cong \...
- 9,870
15
votes
1
answer
389
views
Relations among Pontryagin numbers
The Hattori-Stong theorem describes the image of the morphism:
$$\tau:\Omega^{SO}_*\rightarrow H_*(BSO;\mathbb{Q})$$
that associates to any closed, smooth, oriented manifold $M$ the homology class $\...
- 9,870
15
votes
0
answers
592
views
What is the determinant of Poincaré duality?
For a complex $C^{\bullet}$ of finite dimensional vector spaces, one has a determinant
$$|C^\bullet|:= \bigotimes \left(\Lambda^{top} C^i\right)^{(-1)^i}$$
functorial with respect to quasi-...
- 8,723
14
votes
2
answers
1k
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Converse of Poincaré-Hopf theorem
Let $M$ be a connected, compact, oriented manifold of dimension $n<7$. If any two maps $M \to M$ having equal degrees are homotopic, must $M$ be diffeomorphic to the $n$-sphere?
- 471
14
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2
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1k
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Very particular kind of 4-manifolds. Classification
Let $M$ be a smooth orientable compact connected (with boundary) manifold of dimension $4$. In addition $M$ is assumed to be aspherical and acyclic.
Question: is there a "classification" of ...
- 223
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3
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2k
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Motivation and unsolved problems of TQFT
I have been studying topological quantum field theory by mainly reading the Turaev's book.
I'd like to know if there are unsolved problems that motivate mathematicians to study TQFT, like Riemann's ...
14
votes
7
answers
6k
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The Symmetry of a Soccer Ball
Let $P$ be a polyhedron which satisfies the following three conditions:
$P$ is built out of regular hexagons and regular pentagons.
Three faces meet at each vertex.
$P$ is topologically a sphere.
An ...
- 865
14
votes
2
answers
906
views
Acyclic group and finite CW-complex
Is there a nontrivial example of an acyclic group $G$ such that its corresponding Eilenberg space $K(G,1)$ is homotopy equivalent to a finite CW-complex ?
- 717
14
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3
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1k
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Quotient of solid torus by swapping coordinates on boundary
Let $T$ be the solid 2-torus and let $\sim$ be the equivalence relation on $T$ generated by the relation $\{(\alpha,\beta) \sim (\beta,\alpha) \mid \alpha, \beta \in S^1\}$ on the boundary $\partial T=...
- 1,813
14
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3
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683
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Compact manifolds with big mapping class group
I was wondering if compact surfaces were the only compact manifolds with a "big" or "complicated" mapping class group.
Are there higher dimensional manifolds (which are not in some way reducible to ...
- 2,696
14
votes
2
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2k
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Topological Conformal Field Theories.
Hi there!
My question is simple, and I hope you don't misunderstand it. Why should one care about Topological Conformal Field Theories? What are the motivations to study it? I mean, for example: ...
- 229
14
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3
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991
views
Homotopy type of set of self homotopy-equivalences of a surface
Let $\Sigma$ be an oriented topological surface. For simplicity, assume that the genus of $\Sigma$ is at least $2$. There are a number of classical results on the homotopy types of various groups of ...
- 44.8k
14
votes
2
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829
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Can a cyclic group of prime order act on a rationally acyclic finite dimensional complex and have no fixed points?
Let $C_p$ b the cyclic group of order $p$, with $p$ a prime.
Is is possible for $C_p$ to act (cellularly) on a rationally acyclic finite dimensional CW complex $X$ with no fixed points?
Standard Smith ...
- 11.1k
14
votes
1
answer
826
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An almost complex structure on $S^2\times ...\times S^2 / \mathbb{Z_2}$
Consider the product of $2n$ two-spheres $X_n=(S^2)^{2n}$. This manifold admits an orientation preserving involution that preserves the product structure and acts as the (orientation reversing) ...
- 14.3k
14
votes
2
answers
1k
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Construction of invariants of 4-manifolds with the Kirby calculus
I'm an undergraduate student, interested in the low dimensional topology, in particular, the 4-manifold theory.
I have a question.
In the knot theory, the Reidemeister moves play fundamental roles.
...
14
votes
2
answers
1k
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Atiyah duality without reference to an embedding
Atiyah duality is the equivalence $M/\partial M \simeq (M^{-T(M)})^\vee$, i.e. the Spanier-Whitehead dual of the space $M/\partial M$ is the Thom complex of the stable normal bundle of $M$. The ...
- 5,839
14
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1
answer
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Who proved that two homotopic embeddings of one surface in another are isotopic?
If $\Sigma_1$ and $\Sigma_2$ are two compact topological surfaces with boundary and $\phi, \psi : \Sigma_1 \hookrightarrow \Sigma_2$ are two orientation-preserving embeddings that are homotopic, then ...
- 10.1k
14
votes
2
answers
788
views
Restriction of a branched cover to its branch locus
Assume that we have a smooth, compact, complex surface $X$, and a smooth and irreducible divisor $B \subset X$. Let $G$ be a finite group. For every group epimorphism $$\varphi \colon \pi_1(X-B) \to G,...
- 66.3k
14
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2
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2k
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Applications of homotopy groups of spheres
The study of the homotopy groups of spheres $\pi_i(S^n)$ is a major subject in algebraic topology. One knows for example that nearly all of them are finite groups. Some are explicitly known. There is ...
- 1,085
14
votes
1
answer
933
views
Smooth structures on PL 4-manifolds
Is it known whether $O(4) \to PL(4)$, the map from the orthogonal group to the group of piecewise linear homeomorphisms of $\mathbb{R}^4$, is a homotopy equivalence? By smoothing theory for PL ...
- 967
14
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1
answer
449
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Is there a Morita cocycle for the mapping class group Mod(g,n) when n > 1?
Write Mod(g,n) for the mapping class group of a genus-$g$ surface $\Sigma$ with $n$ boundary components. When $n=0,1$ we define the Torelli group $T$ to be the subgroup of Mod(g,n) which acts ...
- 19.2k
14
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1
answer
681
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Alexander duality for non-manifolds
Let $X$ be a CW complex and $A$ a subcomplex. I will assume that both are compact, and that $X$ is $n$-dimensional. Furthermore, assume that the local homology of $X$ is that of a manifold in a ...
- 4,133
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2
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1k
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Exotic spheres and stable homotopy in all large dimensions?
Given that the kervaire invariant one problem has been solved in (almost) all dimensions....my question is whether there exists an exotic sphere in all sufficently lagre dimensions? Given the Kervaire-...
- 703
14
votes
1
answer
860
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Mapping torus of Klein bottle
This got 5 upvotes but no answers on MSE (Mapping torus of Klein bottle), so I'm cross-posting to MO:
The mapping torus of a Klein bottle $ K $ is a compact flat 3 manifold.
The mapping class group of ...
- 4,081
14
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0
answers
326
views
When can we extend a diffeomorphism from a surface to its neighborhood as identity?
Let $M$ be a closed and simply-connected 4-manifold and let $f: M^4 \to M^4$ be a diffeomorphism such that $f^*: H^*(M;\mathbb{Z})\to H^*(M;\mathbb{Z})$ is the identity map. Moreover, let $\Sigma \...
- 3,751
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0
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225
views
Hauptvermutung for non-manifolds
The Hauptvermutung proposes the following: if two finite simplicial complexes are homeomorphic then they are PL-homeomorphic, meaning that they have a common refinement.
People are mostly interested ...
- 2,050
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0
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338
views
Are there Alexander-Whitney maps in geometric homology?
When $X$ is a smooth manifold, Lipyanskiy defines a chain complex whose homology is isomorphic to singular homology -
let's say $GC_*(X)$ - generated by maps $\sigma: M \to X$ from compact $k$-...
- 9,580
13
votes
6
answers
3k
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Geometric flavored textbook on algebra
I am interested in topology, while I am not so comfortable with some
algebraic flavored textbook on algebra. Actually, it was not until I learned some topology that I began to understand some ...
- 139
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2
answers
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What characteristic class information comes from the 2-torsion of $H^*(BSO(n);Z)$?
This is just a general curiosity question:
In the standard textbook treatments of characteristic classes, and in particular the treatment of universal Pontrjagin classes, it's standard to consider $H^...
- 5,489
13
votes
3
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1k
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Manifold whose universal covering is a sphere but which is not a space form?
Let $M^n$ be a smooth manifold whose universal cover is homeomorphic $\mathbb{S}^n$, are there examples where $M^n$ is not homeomorphic to a space form ?
The answer may vary if you replace ...
- 4,123
13
votes
3
answers
2k
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A quotient space of complex projective space
Let $\mathbb{C}P^n$ be the $n$-dimensional complex projective space and denote $[z_0:\dots:z_n]$ its points. If we glue $[z_0:\dots:z_n]$ and $[\overline{z_0}:\dots:\overline{z_n}]$ for any $[z_0:\...
- 331
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1
answer
552
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Realizing symmetric groups by diffeomorphisms
Let $M$ be a (closed, smooth) manifold of dimension $d$. For $n$ a positive integer, fix $n$ points $x_1, \dots, x_n \in M$. The group of diffeomorphisms of $M$ that permutes the points $x_i$ surjects ...
- 11.9k
13
votes
2
answers
534
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Heegaard splitting of covering hyperbolic manifold.
I am curious about how the Heegaard genus changes after a finite covering.
Is there anyone constructing an closed hyperbolic 3-manifold $N$ such that
the Heegaard genus of a finite covering of $N$ ...
- 841
13
votes
3
answers
1k
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How to disjoint two cycles with zero intersection?
Suppose that $M^n$ is a smooth connected orientable manifold and $Z^k$ with $Z^{n-k}$ are two real cycles in $M^n$ with zero index of intersection $Z^k\cdot Z^{n-k}=0$ (these cycles are submanifolds ...
- 14.3k
13
votes
3
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2k
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Is there a good definition of (topological) K-Theory over arbitrary spaces?
Hi
(this is my very first question here, so please don't hurt me...)
for some time now i've been looking for a sufficiently aesthetical definition of (topological) K-theory of arbitrary spaces, yet ...
- 333
13
votes
2
answers
2k
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Surface bundles over a surface
What can be used to distinguish two $\Sigma_g$-bundles over $\Sigma_h$ up to
(1) homotopy?
(2) homeomorphism?
(3) fiberwise homeomorphism?
(4) bundle isomorphism?
And can these always be computed ...
- 2,734
13
votes
1
answer
580
views
Identifying two definitions of orientation on a vector space
Let $V$ be an $n$-dimensional real vector space. Here are two definitions of an orientation on $V$:
A generator of the $1$-dimensional vector space $\wedge^n V$, up to multiplication by positive ...
- 133
13
votes
1
answer
518
views
Low dimensional homotopy groups of $\operatorname{Top}(4)$
$\DeclareMathOperator\Top{Top}$I would like to compute $\pi_3\Top(4)$ and $\pi_4\Top(4)$. It is known that $\Top(4)/O(4) \rightarrow \Top/O$ is 5-connected and
$$
\pi_k(\Top/O) =
\begin{cases}
...
13
votes
1
answer
506
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The Hopf Invariant 1 Problem through L-polynomials
Adams and Atiyah give a wonderfully simple proof of the Hopf invariant 1 problem that uses the Adams operations on K-theory to reduce the Hopf Invariant 1 question to an elementary number theory ...
- 5,839