This chapter studies cotorsion pairs and abelian model structures on chain complexes over additive and exact categories. Fundamental properties of the chain homotopy relation and contractible chain complexes are developed before it is shown that the category of chain complexes over any additive category is a Frobenius category. This determines an abelian model structure whose homotopy category is the classical chain homotopy category of complexes. We then examine general properties of abelian model structures on chain complexes. Formal (Quillen) homotopy categories of complexes are identified with triangulated subcategories of the classical chain homotopy category. The homotopy categories are also identified with Verdier quotients of the classical chain homotopy category. The end of the chapter constructs abelian models for derived categories. The first construction is an abelian model structure for the derived category of any exact category with coproducts, kernels, and a set of (small) projective generators. Second, it is shown how any complete hereditary cotorsion pair on a Grothendieck category lifts to a hereditary abelian model structure on the associated category of chain complexes.

We introduce the concept of a standard form for two embedded maximal sphere systems in the doubled handlebody, and we prove an existence and uniqueness result. In particular, we show that pairs of maximal sphere systems in the doubled handlebody (up to homeomorphism) bijectively correspond to square complexes satisfying a set of properties. This work is a variant on Hatcher's normal form.