Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

I am reading the paper,ON ATTACHING 3-HANDLES TO A 1-CONNECTED 4-MANIFOLD by BRUCE TRACE here.He says in this paper that we need only construct a knot $K\subset \partial W^4$ which meets $Σ ^2$ transversely in a single point (i.e., $K$ and $Σ ^2$ are complementary in $\partial W ^4$ ) ,where $W^4$ is 1-connected smooth 4-manifold and $Σ^2$ is 2-sphere in $\partial W^4$.I read this and I can't understand this because I think that $K$ have at least two points in which $K$ intersects $\ Σ^2$. If you can understand why we take such a knot $K$ and find the point of my misunderstanding this ,could you teach me the reason for the existence of $K$ and correct me?

share|cite|improve this question

I haven't read the paper, but I assume the sphere $\Sigma^2$ does not bound a $3$-ball, and that the $3$-manifold $\partial W$ is prime: it is not a connected sum of other $3$-manifolds. In this case, the sphere $\Sigma^2$ does not disconnect $\partial W$ into two pieces. So if you take a small transverse arc piercing the $2$-sphere once, the two ends of the arc lie in the same connected component of $\partial W\setminus \Sigma^2$ and can therefore be connected by another arc, which is embedded by transversality.

This happens a lot in $3$-manifolds. The easiest example is $S^1\times S^2$. If $(p,q)\in S^1\times S^2$, then $S^1\times\{q\}$ is a knot intersecting the sphere $\{p\}\times S^2$ exactly once.

share|cite|improve this answer
Sorry,I can't understand the definition of prime so,could you give me some examples of prime set? – Takahiro Oba Jun 30 '12 at 6:46
A connected sum of two manifolds is defined by removing a small open ball from each and gluing them together along their spherical boundaries. In particular any $S^{n-1}$ which is embedded in an $n$-manifold and which separates the manifold into two pieces, indicates that the manifold is a nontrivial connected sum of two simpler manifolds. – Grumpy Parsnip Jun 30 '12 at 7:52
OK,I see.Judging from your answer,you say that we can choose such a $K$ only if $\partial W$ is prime,don't you? – Takahiro Oba Jun 30 '12 at 8:27
If you have a nontrivial embedded sphere in a prime manifold, then there is definitely a knot $K$ which hits it one point. In a non-prime manifold, some spheres may have such a $K$ and some may not. – Grumpy Parsnip Jun 30 '12 at 8:42
OK,thank you :) – Takahiro Oba Jun 30 '12 at 8:50

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.