All Questions
Tagged with gt.geometric-topology at.algebraic-topology
1,145 questions
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
1
vote
2
answers
897
views
Can a sphere be a phase space?
Put in other words, given an even-dimensional sphere $S^{2k}$: is there a manifold $M$ such that $T^* M$ is diffeomorphic to $S^{2k}$?
- 13
1
vote
2
answers
133
views
free complex with mod-p coefficients
How does one prove the following fact. I could not find anything in literature.
Let $\pi$ be a subgroup of the symmetric group $S_p$ and let $W$ be a free $\pi$-complex. Then for any space $X$ there ...
- 471
10
votes
1
answer
489
views
Visualising locally flat embeddings of surfaces in R^4
As far as I understand it follows from the work of M. Freedman that there exist locally flat embeddings of two dimensional surfaces in $\mathbb R^4$ that can not be smoothed in the class of locally ...
- 14.3k
8
votes
5
answers
4k
views
Line bundle on $S^2$
How do you prove that a line bundle (vector bundle of rank 1) on $S^2$ is isomorphic to the trivial line bundle? Can you give a reference?
Thanks.
- 119
5
votes
0
answers
378
views
Eilenberg-Mac Lane spaces for surface group extensions.
(The question has been edited. It was pointed out in the comments that $\Gamma_G$ could be a surface group, thought of as a finite extension of another surface group, in which case $G$ is finite.)
...
- 10.6k
9
votes
4
answers
3k
views
Associativity of topological join and join of spheres
This must be an elementary question, as I couldn't find any proofs on the Internet, but I still can't do it. And yes, Hatcher says that the join is not actually associative for general topological ...
- 1,000
7
votes
1
answer
824
views
Which Abelian Group sequences arise as the Homology of Embedded CW Complexes?
Background
Let $\mathcal{A} = \lbrace A_0, \ldots, A_M \rbrace$ be an arbitrary sequence of finitely generated Abelian groups. It is well-known that a finite CW complex $X_\mathcal{A}$ may be ...
- 15.5k
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
9
votes
3
answers
735
views
Judging whether a finitely presented group is a 3-manifold group?
Given a finitely presented group $G$, how many necessary conditions do people know for $G$ to be isomorphic to the fundamental group of some closed connected 3-manifold? (e.g. residually finite)
1
vote
0
answers
366
views
Question on Steenrod realizability problem
René Thom proved that for any topological space $X$, any integral homology class of $X$ (of any degree) has an odd multiple that is the pushforward of the fundamental class of a smooth compact ...
- 257
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
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 ...
- 5,509
5
votes
1
answer
376
views
Extension of covering map
The question is the following: suppose I have manifolds $N$ and $M$ both with boundary, and I have a covering map $\phi$ from $\partial N$ to $\partial M.$ The question is: when is there a covering ...
- 96.4k
13
votes
3
answers
1k
views
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
41
votes
0
answers
1k
views
Homotopy type of TOP(4)/PL(4)
It is known (e.g. the Kirby-Siebenmann book) that $\mathrm{TOP}(n)/\mathrm{PL}(n)\simeq K({\mathbb Z}/2,3)$ for $n>4$. I believe it is also known (Freedman-Quinn) that $\mathrm{TOP}(4)/\mathrm{PL}(...
- 6,210
9
votes
2
answers
367
views
Is compact flat manifold cusp cross-sections of a complete finite volume hyperbolic manifold?
Let $M^{n-1}$ be a closed flat manifold. Is it true that there exists a hyperbolic manifold $N^n$ with finite volume such that $M$ is a cusp cross-section of $N$?
It was proved in "On the geometric ...
- 2,623
9
votes
2
answers
659
views
Can the fundamental group of an intersection of a homeomorphic image of a ball with a complement of a ball in $R^3$ be perfect?
I have the following problem: Let $A, B\subset R^3$, $A$ is homeomorphic to a ball, while $B$ is a standard Euclidean ball. Can it happen that the fundamental group of $A\setminus B$ is a perfect ...
- 135
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 ...
2
votes
1
answer
377
views
Homotopy Extension Property (HEP)
I want to show (although Artin gave an ad hoc proof) that if two braids \beta and \beta' are isotopic as braids, then they are equivalent as tangles. I'd like to use the homotopy extension property (...
- 21
22
votes
2
answers
1k
views
The image of the point-pushing group in the hyperelliptic representation of the braid group
Let $B_{2g+1}$ be the Artin braid group on $2g+1$ strands. There is a symplectic representation
$\rho: B_{2g+1} \rightarrow Sp_{2g}(\mathbf{Z})$
called the "hyperelliptic representation," which ...
- 19.2k
14
votes
3
answers
2k
views
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 ...
6
votes
1
answer
489
views
Equivariant handle decompositions
Suppose I have some smooth closed high-dimensional manifold $M$ acted on smoothly by a finite group $G$. By a metric averaging procedure, we can equip $M$ with a smooth Riemannian metric so that $G$ ...
- 18.7k
11
votes
4
answers
1k
views
Knot diagrams, sets of moves and equivalence relations
Short version: Does anyone study equivalence classes generated by a given set of "moves" (in the sense of, but not limited to, Reidemeister moves) on the set of knot diagrams?
Yes, I ...
- 17.6k
5
votes
1
answer
1k
views
Create examples of deg=1 maps
My question is:
For given k-dimension manifold M, Is there any way to produce a manifold N so that there exist a map $f:M\rightarrow N$, $degf=1$
The trivial method I know:
The only trivial method I ...
- 703
1
vote
0
answers
365
views
Killing homotopy groups by removing subsets
Let $X$ be a locally finite CW-complex and let $U$ be an open subset of $X$. Given a non-zero homotopy class $x\in\pi_i(U)$ say, is it possible to find a closed subset $Z\subset U$ whose removal from $...
5
votes
1
answer
592
views
Historical question: fiber bundles
I am sorry if this question is too trivial but I couldn't find the answer.
Who did first classify topological principal $G$-bundles for some topological group $G$? So, I mean that equivalence ...
- 471
5
votes
2
answers
1k
views
Intersection forms of 4-manifolds with boundary
Let $X$ be any simply connected smooth 4-manifold with a fixed Euler characteristic $e$, signature $\sigma$ and boundary $Y$. Assume that the determinant of the intersection form $Q_{X}$ is equal to a ...
- 53
1
vote
1
answer
2k
views
Homology and homotopy type for knot complements
I'm trying to understand a paper by Kawauchi and Matumoto, 'An estimate of infinite cyclic coverings and knot theory.' In one of their proofs they have a manifold $E$ which is the complement of a ...
- 1,025
8
votes
2
answers
518
views
High-dimensional ribbon knots
Let us suppose that we have a ribbon embedding $S^n \rightarrow S^{n+2}$ for $n\geq 3$. Call this knot $K$. By a theorem of Levine (and Trotter for $n=3$ I believe) we know $K$ is unknotted if the ...
- 1,025
4
votes
3
answers
713
views
Jordan curve theorem for cylinders
Hello,
I would like to know if the following result is true:
Let $A,B$ be two embedded circles in $S^2$ which do not intersect and let $C$ be the $\textit{closed}$ region bounded by $A$ and $B$ (...
- 41
24
votes
1
answer
2k
views
How to "see" that double suspension of homology 3-sphere is homeomorphic to a sphere?
Is there a good way to think about/understand the result that the double suspension of a homology 3-sphere is homeomorphic to a sphere, to get intuition for why this is true? For instance, what sort ...
- 3,532
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
4
votes
0
answers
576
views
Topological version of two results in smooth Morse theory
Morse theory is generally presented in the DIFF category. However, there is a version of Morse theory in TOP (see the post Morse theory in TOP and PL categories? for references).
It is well known ...
- 2,136
20
votes
2
answers
1k
views
Characteristic classes for block bundles
Block bundles are PL analogs of vector bundles, see e.g. Rourke-Sanderson's
article
in Bulletin of AMS, 1966. There is a classifying space $B\widetilde{PL}_q$ for rank $q\ $ block bundles, which is ...
- 29.1k
9
votes
1
answer
5k
views
Manifolds are paracompact
By Definition, smooth manifolds are assumed to be Hausdorff and to satisfy the second countability axiom.
I have heard (but never seen written) that these assumptions imply paracompactness (and thus ...
- 10.4k
41
votes
1
answer
6k
views
Classification of surfaces and the TOP, DIFF and PL categories for manifolds
A surface is simply a 2-manifold. The classification theorem for compact connected surfaces (with boundary) is commonly regarded in the categories TOP, DIFF and PL. Well known proofs (e.g. via ...
- 2,136
1
vote
2
answers
391
views
Mapping class between coverings of Riemann surfaces
Let $X$ be a closed Riemann surfaces of genus $g$ and let $p_1:Y_1 \rightarrow X$ and $p_2:Y_2 \rightarrow X$ be two $K$-sheeted, connected, unramified coverings of the Riemann surface. By the theorem ...
- 471
10
votes
2
answers
497
views
Equivariant cohomology of the complement to the arrangement $\bigcup_{i\neq j}\vec x_i = \vec x_j$?
$\DeclareMathOperator\SO{SO}\DeclareMathOperator\SU{SU}\DeclareMathOperator\Conf{Conf}$Let $V=\mathbb{R}^d$ be a $d$-dimensional (Euclidean) vector space over real numbers.
Let $G=\SO(V)$ be the ...
- 133
3
votes
1
answer
459
views
When is the Freudenthal compactification an ANR?
Let $X$ be a locally compact metric ANR (or, if preferred, a locally compact simplicial complex). If needed, assume that $X$ has finitely many ends or is of finite dimension. My question is:
What are ...
- 2,104
5
votes
1
answer
232
views
Finite index subgroups of the mapping class group with geometric meaning
I have got a question that is perhaps not precise in a mathematical sense.
Is there a classification of all coverings of the moduli space of Riemann surfaces which are moduli spaces themselves, that ...
- 471
11
votes
2
answers
1k
views
Infinite loop space structure of $BU^+$
It is well-known that $BU^+$ is homotopy equivalent to an infinite loop space where $U$ is the limit of the unitary groups $U(n)$ for $n \rightarrow \infty$ and $+$ denotes Quillen's Plus construction....
- 165
50
votes
4
answers
3k
views
To which extent can one recover a manifold from its group of homeomorphisms
Question. Suppose that $M$ is a closed connected topological manifold and $G$ is its group of homeomorphisms (with compact-open topology). Does $G$ (as a topological group) uniquely determine $M$?
One ...
- 31.2k
19
votes
3
answers
3k
views
When does the tangent bundle of a manifold admit a flat connection?
Let $M$ be a smooth manifold, and let $TM$ denote its tangent bundle. Under what conditions does $TM$ admit a flat connection $\omega$?
Edit: Formerly, I asked about a flat connection on the frame ...
- 8,512
4
votes
2
answers
921
views
Cone over the Join of two topological spaces
Suppose $X$ and $Y$ are topological spaces. Let's define the join $X\ast Y$ as the quotient space $X\times Y\times [0,1]/\sim$, where $\sim$ is the equivalence relation generated by $(x,y,0)\sim(x,y',...
- 45
8
votes
3
answers
3k
views
Where can I find a full proof of the Chern-Gauss-Bonnet theorem ?
Hello,
I am looking for a proof for the Chern-Gauss-Bonnet theorem. All I have found so far that I find satisfactory is a proof that the euler class defined via Chern-Weil theory is equal to the ...
- 83
9
votes
4
answers
3k
views
Poincaré dodecahedron space
The Poincaré homology sphere $X$ is constructed by identifying opposite faces of a dodecahedron by a (clockwise) twist of 36 degree.
Many books say its fundamental group $\pi_1(X)$ is the binary ...
- 133
3
votes
1
answer
828
views
Moise's Theorem and the Fundamental Domain of a $3$-Manifold
I'm currently researching the relationship between Moise's Theorem (Every closed $3$-manifold has a triangulation) and other properties of manifolds. In particular, I'm trying to learn about the ...
- 1,431
5
votes
2
answers
399
views
whether a kind of surgery can go on infinitely many steps?
let $M$ be a closed ortientable irreducible 3-mfd, let $T$ be a non-separating torus in $M$, we cut $M$ along $T$ and glue two solid tori along the two boundary tori, we get a new closed 3-mfd $M_1$ (...
- 336
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