Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Disk complexes and genus two Heegaard splittings for non-prime
  3-manifolds   Given a genus two Heegaard splitting for a non-prime 3-manifold, we define a
special subcomplex of the disk complex for one of the handlebodies of the
splitting, and then show that it is contractible. As applications, first we
show that the complex of Haken spheres for the splitting is contractible, which
refines the results of Lei and Lei-Zhang. Secondly, we classify all the genus
two Heegaard splittings for non-prime 3-manifolds, which is a generalization of
the result of Montesinos-Safont. Finally, we show that the mapping class group
of the splitting, called the Goeritz group, is finitely presented by giving its
explicit presentation.

8
The categories of this paper are: hep-ex
34
The categories of this paper are: chem-ph
27
The categories of this paper are: comp-gas
15
The categories of this paper are: nucl-ex
30
The categories of this paper are: q-alg
7
The categories of this paper are: hep-th
11
The categories of this paper are: hep-lat
37
The categories of this paper are: supr-con
13
The categories of this paper are: quant-ph
9
The categories of this paper are: nlin
29
The categories of this paper are: funct-an
25
The categories of this paper are: patt-sol
32
The categories of this paper are: atom-ph
23
The categories of this paper are: solv-int
22
The categories of this paper are: chao-dyn
1
The categories of this paper are: cs