From Rotman's Algebraic Topology in a section talking about finitely based chain complexes; I have come across this page and am confused by some of the notation and words.
$(1)$ What is the adequate subcomplex here, are they saying the identified torus of the square is the adequate subcomplex?
$(2)$ I understand that chains in a complex $E_*$ are the elements of each $E_q$, but what does $E_2=\langle P \rangle$ mean? What does $E_1=\langle a \rangle \oplus \langle b \rangle$ mean?
$(3)$ And how do these differentiations (boundary operator) imply the bottom results?
