39 |
39 |
40 Throughout, we have resisted the temptation to work in the greatest possible generality. |
40 Throughout, we have resisted the temptation to work in the greatest possible generality. |
41 %(Don't worry, it wasn't that hard.) |
41 %(Don't worry, it wasn't that hard.) |
42 In most of the places where we say ``set" or ``vector space", any symmetric monoidal category |
42 In most of the places where we say ``set" or ``vector space", any symmetric monoidal category |
43 with sufficient limits and colimits would do. |
43 with sufficient limits and colimits would do. |
44 We could also replace many of our chain complexes with topological spaces (or indeed, work at the generality of model categories). |
44 Similarly, in many places chain complexes could be replaced by more general objects, but we have not pursued this. |
45 |
45 |
46 {\bf Note:} For simplicity, we will assume that all manifolds are unoriented and piecewise linear, unless stated otherwise. |
46 {\bf Note:} For simplicity, we will assume that all manifolds are unoriented and piecewise linear, unless stated otherwise. |
47 In fact, all the results in this paper also hold for smooth manifolds, |
47 In fact, all the results in this paper also hold for smooth manifolds, |
48 as well as manifolds equipped with an orientation, spin structure, or $\mathrm{Pin}_\pm$ structure. |
48 as well as manifolds (PL or smooth) equipped with an orientation, spin structure, or $\mathrm{Pin}_\pm$ structure. |
49 We will use ``homeomorphism" as a shorthand for ``piecewise linear homeomorphism". |
49 We will use ``homeomorphism" as a shorthand for ``piecewise linear homeomorphism". |
50 The reader could also interpret ``homeomorphism" to mean an isomorphism in whatever category of manifolds we happen to |
50 The reader could also interpret ``homeomorphism" to mean an isomorphism in whatever category of manifolds we happen to |
51 be working in (e.g.\ spin piecewise linear, oriented smooth, etc.). |
51 be working in (e.g.\ spin piecewise linear, oriented smooth, etc.). |
52 In the smooth case there are additional technical details concerning corners and gluing |
52 In the smooth case there are additional technical details concerning corners and gluing |
53 which we have omitted, since |
53 which we have omitted, since |