Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Virtual Betti numbers of real algebraic varieties   The weak factorization theorem for birational maps is used to prove that for
all nonnegative i the ith mod 2 Betti number of compact nonsingular real
algebraic varieties has a unique extension to a "virtual Betti number" beta_i
defined for all real algebraic varieties, such that if Y is a closed subvariety
of X, then beta_i(X) = beta_i(X\Y) + beta_i(Y).

19
The categories of this paper are: eess
4
The categories of this paper are: cond-mat
20
The categories of this paper are: acc-phys
26
The categories of this paper are: bayes-an
6
The categories of this paper are: astro-ph
31
The categories of this paper are: ao-sci
28
The categories of this paper are: alg-geom
11
The categories of this paper are: hep-lat
5
The categories of this paper are: gr-qc
23
The categories of this paper are: solv-int
3
The categories of this paper are: physics
22
The categories of this paper are: chao-dyn
37
The categories of this paper are: supr-con
12
The categories of this paper are: nucl-th
32
The categories of this paper are: atom-ph
10
The categories of this paper are: q-bio