Handle decomposition of compact orientable 4-manifolds

Biplab Basak, Manisha Binjola

Published 2021-03-25Version 1

In this article we study a particular class of compact connected orientable PL $4$-manifolds with empty or connected boundary and prove the existence of each handle in its handle decomposition. We particularly work on the compact connected orientable PL $4$-manifolds with rank of fundamental group to be one. Our main result is that if $M$ is a closed connected orientable $4$-manifold then $M$ has either of the following handle decompositions: (i) one $0$-handle, two $1$-handles, $1+\beta_2(M)$ $2$-handles, one $3$-handle and one $4$-handle, (ii) one $0$-handle, one $1$-handle, $\beta_2(M)$ $2$-handles, one $3$-handle and one $4$-handle, where $\beta_2(M)$ denotes the second Betti number of manifold $M$ with $\mathbb{Z}$ coefficients. Further, we extend this result to any compact connected orientable $4$-manifold $M$ with boundary and give three possible representations of $M$ in terms of handles.