In this chapter we prove the Blakers–Massey Theorem and its dual for n-dimensional cubes, which encompass and generalize Theorems 4.2.1 and 4.2.2. We will focus on the three-dimensional case before presenting the complete n-dimensional result. This three-dimensional case, which is much easier to digest than the general case, contains most of the main ideas of the proofs.
As with Theorem 4.2.1, there is a geometric step for which we will provide two proofs – one based on transversality (due to Goodwillie [Goo92]), and the other using purely homotopy-theoretic methods (due to the first author [Mun14]). The latter version replaces the “dimension counting” transversality argument with a “coordinate counting” one. There is also a second formal step which relies on general facts about connectivities of maps and (co)cartesian cubes. Our proof of the formal step is organized differently than in Goodwillie's original work, but it is otherwise very close. The spirit of this part of the proof is best captured by the three-dimensional case.
Before looking at this section, the reader might want to glance at Section 4.1 for the history of the original Blakers–Massey Theorem for triads and the variant in terms of squares.
The general Blakers–Massey Theorem, also known as the Blakers–Massey Theorem for n-cubes or (n + 1)-ad Connectivity Theorem is essentially a statement about higher-order excision for homotopy groups. Some of its far-reaching consequences will be recounted in Section 6.2 and in Chapter 10.
In its early form, the statement concerns the homotopy groups and the connectivity of an (n + 1)-ad (X; X1, …, Xn) which are defined analogously to that of a triad (see Section 4.1). For details, see [BW56, Section 2].
The first proof of this theorem is due to Barratt and Whitehead [BW56] although the hypotheses there require to be simply-connected and ki ≥ 2. The improvement stated above is due to Ellis and Steiner [ES87] who generalize techniques of Brown and Loday [BL87a].
As in the classical Blakers–Massey case, of special importance is also the first non-vanishing homotopy group of the (n + 1)-ad, which Ellis and Steiner [ES87], using work of Brown and Loday [BL87b], show to be given by an isomorphism