5
votes
3answers
80 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 …
1
vote
4answers
166 views
circle action on sphere
surely $S^1$ can act on $S^n$ as a rotation.I want to know if there is some other way that a circle can act on sphere.
13
votes
3answers
363 views
Fundamental groups of noncompact surfaces
I got fantastic answers to my previous question (about modern references for the fact that surfaces can be triangulated), so I thought I'd ask a related question. A basic fact abo …
2
votes
1answer
219 views
K-theory as a generalized cohomology theory
Which of the statements is wrong:
a generalized cohomology theory (on well behaved topological spaces) is determined by its values on a point
reduced complex $K$-theory $\tilde K …
6
votes
2answers
160 views
Relation between $KO$ and $K$
What can be said about the relation between the complex and the real K-theory of a CW complex? An $n$-dimensional complex vector bundle is an $2n$-dimensional real vector bundle bu …
18
votes
4answers
558 views
Is the universal covering of an open subset of $\mathbb{R}^n$ diffeomorphic to an open subset of $\mathbb{R}^n$ ?
Is the universal covering of a connected open subset $U$ of ℝn diffeomorphic to an open subset of ℝn (standard differentiable structure)?
If not true in general, is t …
3
votes
1answer
212 views
motivation of surgery
an $n$-surgery on m dim manifold M is to cut out $S^n\times D^{m-n}$and replace it by $D^{n+1}\times S^{m-n-1}$.
I want to know how this is invented?
I do know that the effect of p …
6
votes
1answer
183 views
Seiberg-Witten theory on 4-manifolds with boundary
What generalizations of Seiberg-Witten theory to 4-manifolds with boundary do exist?
I would be especially interested in theories which "behave good" under gluing along the bounda …
6
votes
5answers
315 views
Geometric group theory and analysis
Geometric group theory is mainly concerned with topological and geometric properties of groups, spaces on which they act etc., so the ideas employed in GGT are mainly algebraic/geo …
3
votes
1answer
206 views
Poincare duality
I am reading the proof of Theorem Poincare duality in "princibles in AG" of Griffith.
They constructed "dual cell decomposition" of a polyhedra decomposition of manifold M and the …
7
votes
3answers
248 views
Triangulating surfaces
I've had a few undergraduate students ask me for references for the classical fact (due to Rado) that closed topological surfaces can be triangulated. I know two sources for this, …
7
votes
2answers
176 views
existence of a connected set with given connected projections.
Suppose A and B are compact connected sets in the XY plane and XZ plane respectively in R^3. Suppose further that the the range of x-values taken by A and B are the same (i.e, proj …
6
votes
2answers
128 views
Closed hyperbolic manifold with right-angled fundamental domain
What is an example (as simple as possible, please!) of a closed hyperbolic three-manifold with a right-angled polyhedron as fundamental domain?
If we allow cusps then the Whi …
12
votes
2answers
336 views
Proofs of Kirby’s theorem
Each orientable 3-manifold can be obtained by doing surgery along a framed link in the 3-sphere. Kirby's theorem says that two framed links give homeomorphic manifolds if and only …
0
votes
0answers
224 views
Again about Bing’s house with two rooms [closed]
Possible Duplicate:
How to show that the “bing’s house with two rooms” is contractible?
I don't know why my question is closed? here, I make my question …
