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0. BASIC HOMOTOPY NOTIONS 5 p^{t e Jf1'2.--"} : ^{1} D a'}. Then M(X)(a) n M(X)(f3) is defined by t _ 1 {l } D a' U /?' = (a n /?)', so equals M(X)(a n /?). Thus we have a topological (n + l)-ad. There is a canonical inclusion X(a) C M(X)(a): we define the t- coordinate (uniquely) by t(a) = 0, t{a') — 1. Moreover, in the diagram valid for i : a C /?, X(a) ^ M ( a ) X(/3)C ^M([3) there is a canonical homotopy making the diagram commute: viz. leave the in- coordinate and the coordinates in a U (3' fixed, but let each coordinate in /? — a have value t at stage £. In the case where X takes values in CW complexes and cellular maps, it is easily verified by induction that M(X) is a CW (n + l)-ad. Finally, X(a) is a deformation retract of M(a): we deform the coordinates in a linearly to 0. In the case of finite CW complexes, this is even a cellular collapse, so we have simple homotopy equivalences. The above construction permits us to define homology, homotopy etc. of topo- logical objects of type 2n ( = functors from 2 n to topological spaces) by con- sidering only topological (n + l)-ads. Our definitions will be self-consistent, for if X is already a topological (n + l)-ad, we have a well-defined projection pi : |M(X)| — • \X\ with each inverse image a cube, and M(X)(a) C p^ 1 (X(a)) a homotopy equivalence. Thus in the CW case, p\ is a homotopy equivalence of (n + l)-ads in general, it is a singular homotopy equivalence. We define the homology of a CW (n + l)-ad K by chain groups: we take the chains of \K\ modulo the union (denoted by \dK\) of all K(a) with a a proper subset of { 1 , . . . , n}. For this we can use any coefficient module over the group ring of 7Ti(|.ftT|), or analogously if \K\ is not connected. The short exact sequences of chain complexes 0 - C*{diK) - C^K) - C*(K) - 0 induce the usual homology exact sequences of the (n + l)-ad. Analogous ob- servations apply to cohomology for a topological (n + l)-ad we use singular chains. Now let K be an (n + l)-ad in the category of based topological spaces. Define F(K) as the function space of all maps J{l,.-,n} __ | K | g u c h t h a t ja x QO/ _ K(a) for each a C { 1 , . . . , n}, and all proper faces with some coordinate 1 map to the base point. Of course, we give F(K) the compact open topology, and base point the trivial map. Now for r 0, define 7rn+r(K) = nr(F(K)) : it is a group for r ^ 1, abelian for r ^ 2. The face operator di induces (compose with I6i) a projection F(K) — F(diK). This is a fibre map: the fibre is the subspace of maps sending the zth face to a point. This is just FfaSiK). We observe that this is the loop space (with