Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Open book decompositions versus prime factorizations of closed, oriented
  3-manifolds   Let $M$ be a closed, oriented, connected 3--manifold and $(B,\pi)$ an open
book decomposition on $M$ with page $\Sigma$ and monodromy $\varphi$. It is
easy to see that the first Betti number of $\Sigma$ is bounded below by the
number of $S^2\times S^1$--factors in the prime factorization of $M$. Our main
result is that equality is realized if and only if $\varphi$ is trivial and $M$
is a connected sum of $S^2\times S^1$'s. We also give some applications of our
main result, such as a new proof of the result by Birman and Menasco that if
the closure of a braid with $n$ strands is the unlink with $n$ components then
the braid is trivial.

12
The categories of this paper are: nucl-th
20
The categories of this paper are: acc-phys
21
The categories of this paper are: adap-org
16
The categories of this paper are: stat
30
The categories of this paper are: q-alg
14
The categories of this paper are: math-ph
27
The categories of this paper are: comp-gas
3
The categories of this paper are: physics
25
The categories of this paper are: patt-sol
31
The categories of this paper are: ao-sci
19
The categories of this paper are: eess
32
The categories of this paper are: atom-ph
29
The categories of this paper are: funct-an
7
The categories of this paper are: hep-th
4
The categories of this paper are: cond-mat
6
The categories of this paper are: astro-ph