Can the the monodromy matrix of the action of a pseudo-Anosov homeomorphism of a surface on it's homology be of the form of a reducible or block matrix? Any such reference can be helpful. Thanks.
Every symplectic matrix arises as the monodromy of a pseudo-Anosov homeomorphism. The canonical reference is Farb-Margalit:
Farb, Benson; Margalit, Dan, A primer on mapping class groups, Princeton Mathematical Series. Princeton, NJ: Princeton University Press (ISBN 978-0-691-14794-9/hbk; 978-1-400-83904-9/ebook). xiv, 492 p. (2011). ZBL1245.57002.
Where you can find the facts that there exists pseudo-Anosovs in the Torelli group (trivial monodromy), and the map from the mapping class group to the Symplectic group is surjective. Taking some element with your favorite matrix, and multiplying it by a high power of a Torelli pseudo-anosov will give you what you want.