p^{t e
: ^{1} D a'}. Then M(X)(a) n M(X)(f3) is defined by
_ 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^
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
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} __ |
g u c h t h a t
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
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
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