All Questions
Tagged with surgery-theory smooth-manifolds
16 questions
23
votes
2
answers
881
views
Vanishing of characteristic numbers vs vanishing of characteristic classes
A famous result by Thom states that Oriented Bordism classes are determined by characteristic numbers; specifically, two closed manifolds are orientedly bordant if and only if they have the same ...
- 521
18
votes
1
answer
565
views
A search for a sequence of $6$-manifolds
How to construct closed, orientable, smooth, simply-connected $6$-manifolds such that $H^{*}(M,\mathbb{Z}) \cong \mathbb{Z}[a]/(a^{4})$ (Where $a$ is a generator of degree 2) satisfying $p_{1}(M) = n ...
- 6,995
17
votes
1
answer
612
views
Is there a notion of a chain complex with corners?
Roughly speaking, algebraic topology works by reducing questions about topological objects such as manifolds and cell to questions about chain complexes.
On the topological side, although in the PL ...
- 22.2k
16
votes
2
answers
2k
views
Smooth structures on the connected sum of a manifold with an Exotic sphere
What can we say about the connected sum of a manifold $M^n$ with an Exotic sphere? Is is possible some of them are still diffemorphic to $M^n$. Is it possible to classifying all the exotic smooth ...
- 1,598
7
votes
3
answers
628
views
Generalized Schoenflies - formalizing step in proof?
[Sorry if the level here is wrong, I asked this on math.SE, but even with a bounty, it got no attention.]
I am currently reading Hatcher's 3-Manifolds notes, the part proving Alexander's theorem, ...
- 174
6
votes
0
answers
197
views
Regarding a proof in the surgery theorem by Gromov and Lawson
I have a question regarding a proof in the article The classification of simply connected manifolds of positive scalar curvature written by Gromov and Lawson. The precise reference is:
Gromov, ...
- 386
6
votes
0
answers
88
views
Surgery on $M\times S^1$
I've encountered such a question and I don't know if it's trivial or not.
Given a simply connected closed $n$-manifold $M$ ($n\geq 4$), $\pi_1(M\times S^1)\cong\mathbb{Z}$ and it can be killed by ...
- 123
4
votes
0
answers
202
views
Possible Euler characteristics of manifolds with tangential structures
Let $p:B\to BO$ be a fibration. We say that a manifold has a $B$-structure if its stable tangent bundle lifts to $B$. I am interested in the question of whether there exists, for a given even ...
4
votes
0
answers
116
views
Finding inverses of certain elements in the set of normal invariants of a smooth manifold
Let, $V$ denote the Stiefel manifold of 2-frames $V_{10,2}$ . Consider the the map $S_\text{diff} (V) \xrightarrow{\eta} N_\text{diff} (V) $ in the surgery exact sequence of a smooth manifold. . ...
- 141
4
votes
0
answers
377
views
Kirby's theorem for 4-manifolds
In dimension 3, we have the celebrated Kirby theorem: Let $L_1, L_2$ be two links in the 3-sphere $S^3$; then they surgeries along them produce homeomorphic 3-manifolds if and only if they are related ...
- 5,230
4
votes
0
answers
249
views
Being a product - from homology to topology
The famous Kunneth formula expresses the homology of a product manifold as the tensor product of the two algebras.
Now suppose we know that a manifold $X$ has a decomposition $H_*(X) \simeq A \otimes ...
- 2,152
3
votes
0
answers
80
views
Quartic link in a 5-sphere
In this post I would like to propose a quartic link in a 5-sphere.
Let us start with the following gluing into a 5-sphere:
$$S^5=(D^2_{} \times T^3_{}) \cup_{T^4} ({S^5 \smallsetminus D^2 \times T^3})...
- 10.5k
2
votes
0
answers
117
views
About connected cobordism and surgery
I need to find various ways of performing two surgeries on a collection of circles so that the resulting 2-dimensional cobordism (the trace of the surgeries) is connected.
How can I find these ? up ...
- 119
2
votes
0
answers
102
views
Realizing an amalgamated product of groups by splitting a closed manifold along a codimension 1 submanifold
In the paper "A splitting theorem for manifolds" by S.E. Cappell,
https://www.maths.ed.ac.uk/~v1ranick/papers/capsplit.pdf
the following "inverse" of the Seifert-van Kampen theorem for closed ...
- 21
2
votes
0
answers
170
views
Triple link in a 5-sphere -- Proposal
In this post I would like to propose a triple link in a 5-sphere.
Let us start with the following gluing into a 5-sphere:
$$S^5=(D^2_{} \times T^3_{}) \cup_{T^4} ({S^5 \smallsetminus D^2 \times T^3})$...
- 10.5k
1
vote
0
answers
169
views
Confused about A. Kosinski's description about surgery in his book "differential manifolds"
Please excuse me, if MO is not the proper place for this question. I aksed the same question on M.SE
https://math.stackexchange.com/questions/3511134/confused-about-a-kosinskis-description-of-surgery-...
- 173