All Questions
49
questions
2
votes
0
answers
133
views
Explicit S-duality map
$\DeclareMathOperator{\Th}{Th}$
The Thom space of a closed manifold $M$ ($\Th(M)$) is the $S$-dual to $M_+ (=M \cup \{pt\})$. Let $M$ be embedded in $\mathbf R^{n+k}$. I found the duality map from $S^...
8
votes
0
answers
144
views
Generators for unstable cobordism
I am looking for explicit descriptions of generators of some low-dimensional unstable cobordism groups. For example, $\mathbb CP^2$ embeds into $\mathbb R^7$ by a result of Haefliger. Because it has ...
- 6,701
1
vote
1
answer
238
views
Reference request and prerequisites for understanding the Sphere Theorem and the Loop Theorem in 3-manifold theory
As part of my directed studies project, my advisor has suggested that I completely understand the proof of the Sphere Theorem and the Loop Theorem in 3-manifold theory and explain it to him. I have ...
- 131
6
votes
1
answer
561
views
Geometric interpretation of transfer map on homology
Let $f\colon M\to N$ a smooth surjective map of compact oriented manifolds of the same dimension. Then there is a map $f_!\colon H_i(N)\to H_i(M)$ obtained from the induced map on cohomology combined ...
- 2,927
3
votes
1
answer
221
views
Example of a non $\pi_1$-injective, degree one, self-map of a three-manifold
All manifolds will be assumed to be closed, oriented, and connected.
Let $f\colon M\to M$ be a map of degree $\pm 1$. It is not hard to show that $\pi_1(f)$ is surjective.
What is an example of a non ...
- 1,097
2
votes
1
answer
320
views
Calculating degree via homotopy
I'm looking for a reference for the following:
Suppose that $f_1,f_2\colon S^n\rightarrow S^n$ are smooth maps. Let $i\colon S^n\rightarrow \mathbb{R}^{n+1}$ be the inclusion, and suppose that $F\...
- 23
5
votes
0
answers
92
views
Equivariant imbedding of compact manifold
Let $G$ be a compact Lie group smoothly acting on a smooth compact manifold $X$.
Is it true that there exists a smooth $G$-equivariant imbedding of $X$ into a Euclidean space acted linearly (and ...
- 21.3k
1
vote
0
answers
131
views
definition of generic function
what is definition of generic function in following paper ? i need a reference for definition generic function .
"A. Hatcher, W. Thurston, A presentation for the mapping class group of a closed ...
- 119
40
votes
2
answers
2k
views
Can the nth projective space be covered by n charts?
That is, is there an open cover of $\mathbb{R}P^n$ by $n$ sets homeomorphic to $\mathbb{R}^n$?
I came up with this question a few years ago and I´ve thought about it from time to time, but I haven´t ...
- 10.4k
8
votes
1
answer
408
views
Orbifolds are Thom-Mather stratified spaces
Where can I find a proof of (or if it is even true) that an (effective) orbifold is a Thom-Mather stratified space?
edit: after some search, I found the proof should be contained in either
GIBSON, C....
- 793
5
votes
0
answers
353
views
CW complex vs analytic manifold vs variety
I am looking to gain some intuition into the passage (or obstruction thereof) between different categories of objects one encounters in geometry and topology. To oversimplify things a bit, the ...
- 1,377
16
votes
0
answers
220
views
Reference request: Milnor rank of spheres
Milnor defines the rank of a smooth manifold $M$ as the maximum cardinality of a linearly independent set of vector fields on $M$ whose elements are pair wise commuting. In other words, the rank of $M$...
- 343
8
votes
1
answer
429
views
On the state of the art on closed $(n-1)$-connected $2n$ manifolds
In the paper "Classification of $(n - 1)$-Connected $2n$-Manifolds" by C.T.C.Wall (Annals of Mathematics , Jan., 1962, Second Series, Vol. 75, No. 1 (Jan., 1962), pp. 163-189), Wall studies ...
- 6,975
15
votes
6
answers
2k
views
Any shortcuts to understanding the properties of the Riemannian manifolds which are used in the books on algebraic topology
I'm now attending a reading seminar on the algebraic topology.
The seminar treats the book of Bott & Tu (Differential Forms in Algebraic Topology) and Milnor (Characteristic Classes).
In those ...
- 1,073
4
votes
0
answers
140
views
Characteristic classes of quotient manifold
Let $M$ be a compact oriented smooth manifold with boundary and let $G$ be a compact Lie group acting smoothly, orientation-preservingly and freely on $M$.
(Under what conditions) is there a ...
- 641