All Questions
Tagged with surgery-theory differential-topology
16 questions
1
vote
0
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161
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Higher dimensional Seifert surfaces and link numbers of higher knots
In 3-manifold topology, the notion of Seifert surface is well known. It is then used to define link numbers of knots.
Question: Consider embedding $N^n \rightarrow M^{2n+1}$ of n-dimensional manifold $...
- 593
4
votes
1
answer
296
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On the proof of the surgery step in Wall's book
This question regards a part of the proof of the so called surgery step, in Wall's book "surgery on compact manifolds", Theorem 1.1.
Setting
$M^m$ smooth manifold, $X$ CW complex, $\phi :M\...
- 2,533
8
votes
1
answer
474
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Regarding the surgery construction in "A procedure for killing homotopy groups of differentiable manifolds" by Milnor
In the first section of "A procedure for killing homotopy groups of differentiable manifolds", Milnor gives the surgery construction as follows. Let $W$ be an $n=p+q+1$ dimensional manifold. ...
- 147
3
votes
2
answers
199
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Effect of a Lutz twist on Euler number
I already asked this question on the Math Stack Exchange but did not get an answer.
I am currently working through Geiges proof of the Martinet-Lutz theorem, which can be
found here, and am trying to ...
6
votes
1
answer
287
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Dehn surgery along primitive knot in 3-dimensional handlebody
I'm studying the article "An alternative proof of Lickorish–Wallace theorem" (doi link)
and I got stuck in a problem.
Let $H_g$ be a 3 dimensional handlebody of genus $g$, a primate curve in ...
- 173
7
votes
3
answers
249
views
Normal invariants
I am having a hard time finding examples of computations of normal invariants of surgery theory (or more generally the set of homotopy classes of maps $[X,G/O]$). Does anybody have good references?
- 641
1
vote
0
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169
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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
7
votes
3
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628
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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
1
answer
170
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Diffeomorphism type of the added sphere in simply connected surgery
A classical result of simply connected surgery theory, is that if two normal maps $f:M_i\rightarrow X$, $i=0,1$ are normally cobordant and if the dimension of the manifolds is odd, there exists a ...
- 93
6
votes
0
answers
180
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Cell structures of simply-connected 5-manifolds (classified by Barden's 1965 paper)
In Barden's 1965 paper: Simply-connected five manifolds, Barden gave a complete list of diffeomorphism classes of simply-connected 5-manifolds:
$$X_{j,k_1,\dots,k_n}=X_j\#M_{k_1}\#\cdots\#M_{k_n}$$
...
- 1,329
18
votes
1
answer
565
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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
2
votes
0
answers
109
views
Compare two topologies: Three 2-tori inside $S^3 \times S^1 \# S^2 \times S^2$ glued from two different diffeomorphisms
We like to ask for the comparison of two topologies of three 2-tori inside the same 4-manifolds glued from two different diffeomorphisms (see the end).
Given an embedded torus $T$ with trivial normal ...
- 10.5k
2
votes
1
answer
166
views
What is the most symmetric configuration of four 2-surfaces linked in $S^4$?
What are some of the most symmetric configurations of four 2-surfaces linked in the 4-dimensional sphere $S^4$?
To make a lower-dimensional analogy, recall that in 3-dimensional sphere $S^3$, we can ...
- 10.5k
1
vote
0
answers
76
views
Does a homotopy sphere that bounds a highly connected manifold also bound a parallelizable manifold?
Suppose that the homotopy sphere $\Sigma^{n}$ can be realized as the boundary of a smooth $(n+1)$-dimensional cobordism that is $(n-1)/2$-connected for $n$ odd (respectively, $(n-2)/2$-connected for $...
- 165
1
vote
1
answer
129
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Connected representant of a framed cobordism class (reference needed)
Let $N^n\subseteq M^m$ be a submanifold with a framing of the normal bundle, $2n<m$. Then $N^n$ is framed cobordant (in $M^m$) to something connected.
I believe it could be proved by directly ...
- 972
16
votes
2
answers
2k
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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