2
votes
2answers
183 views

Elementary question about Isotopy (in the definition of a Teichmuller space)

Disclaimer - I don't have much experience in topology/complex geometry, so I apologize if what I'm asking is too elementary for this site. Let $S$ be some orientable surface obtained by removing ...
4
votes
1answer
223 views

Topological characterisation for a (closed irreducible) hyperbolic 3-manifold

Is there a topological characterisation of what a (closed irreducible) hyperbolic 3-manifold is? I don't know any Riemannian geometry and still want to understand what an exceptional Dehn surgery is. ...
0
votes
1answer
190 views

Fibration in the 3 torus.

The Hopf fibration $S^1\rightarrow S^3\rightarrow S^2$ gives a decomposition of $S^3$ into 2-tori and to circles, so that the tori are foliated by circles of slope 1. If you take the region between ...
5
votes
1answer
265 views

Does there exist a space X whose suspension is homotopy equivalent to [0,1] rel ends but where X is not contractible?

As pointed out by David White in when mapping cone is contractible there exists an acyclic CW-complex $X$ which is not contractible but whose suspension is contractible. Namely, let $a$ and $b$ be ...
3
votes
4answers
367 views

Picturing a Certain Torus and Klein Bottle

The other day I was explaining orientability to someone and we were walking through some of the statements about orientability on the Wikipedia page on the topic. While I was able to satisfy his ...
8
votes
0answers
235 views

Can a composition with itself of a universal self-map be non-universal?

I have formulated (and published) the notion of a universal map (and of a universal morphism), and the problems below, in the early 1960-ies. DEFINITION   A continuous map   $u: ...
2
votes
1answer
242 views

Homotopy groups of K3

Let X be a K3 surface and $Y=X/\mathbb{Z}_2$, an Enrique surface. Long exact sequence of homotopy groups corresponding to fiberaion $\pi:X\to Y$, says that $\pi_2(X)=\pi_2(Y)$, while we know $H_2(X)$ ...
4
votes
0answers
290 views

Calculate the cohomology ring using Morse Theory [closed]

My question is: Is there any way to calculate the cohomology ring of a finite dimension manifold using the Morse theory? The cohomology ring structure strongly rely on the following things ...
0
votes
1answer
231 views

How many 3-flip Mobius strip knots are there?

Take a clock-wise 3-flip mobius strip, Cut it down the middle and then let the ribbon cross itself 6 times. This forms a framed knot of which there are many. Get the knot diagram. I've found ...
3
votes
0answers
135 views

Topology of K3 as a sum of two abelian fibrations.

Let $E$ be a blow-up of $\mathbb{P}^2$ at 9-points in the bases locus of a pencil of elliptic curves (A $T^2$ fibration over $S^2$). K3 surfaces is obtained by removing a fiber from two copies of $E$ ...
4
votes
2answers
457 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 ...
-1
votes
1answer
420 views

Is this manifold orientable? [closed]

Let $C$ be the set of points $(a,b,c,d) \in \mathbb{C}^4$ which satisfy 1) $ \left|a\right|^2+\left|c\right|^2=\left|b\right|^2+\left|d\right|^2 =1 $. 2) $ a\bar{b}+c\bar{d}=0 $ There is a ...
12
votes
5answers
1k views

Is every real n-manifold isomorphic to a quotient of $\mathbb{R}^n$?

I'm curious about the following: Is every real n-manifold isomorphic to a quotient of $\mathbb{R}^n$? Thanks. EDIT: As Tilman points out, the manifold should be connected. Also, yes, I'm thinking ...