Cellular chains of a product of finite CW complexes ($d(e^p\times f^q)=d{e}^p\times f^q+(-1{)}^p{e}^p\times d{f}^q$)

The product cells of finite CW complexes give the tensor product of their cellular chain complexes. The boundary formula is the displayed signed product rule. The sign comes from moving the outward normal of the second factor past the $p$ oriented tangent directions of the first factor. This is the cellular counterpart of the Eilenberg–Zilber theorem.