We study the stable classification of closed connected oriented spin smooth 4-manifolds by using techniques of Kervaire-Milnor surgery. Then we reproduce a nice result of Kurazono and Matumoto [I. Kurazono, T. Matumoto, Hiroshima Math. J. 28, 1998] for such manifolds under the assumption that the fundamental group is finitely presentable and has vanishing second and third homology with Z_2-coefficients.
On the stable classification of spin four-manifolds / Spaggiari, Fulvia. - In: OSAKA JOURNAL OF MATHEMATICS. - ISSN 0030-6126. - STAMPA. - 40:(2003), pp. 835-843.