k-string links f and g have link-homotopic closures if and only if there is a braid-like automorphism 6 of RF(2k) such that we have commutative diagrams

By Theorem 9 we have commutative diagrams as stated, where [theta] = [bar.[beta]].

Given a k-string link f, the n-level group diagram for f, n [greater than or equal to] 2, is the commutative diagram

In order for C to be associative, R must satisfy two pentagonal commutative diagrams, equationally given by

(Coalgebra (B, [DELTA], [epsilon]) is defined dually by coassociative comultiplication [DELTA] : B [right arrow] B [cross product] B and counit [epsilon] : B [right arrow] 1M satisfying the counit diagrams.) That [B.sup.[cross product] (n)]) | [B.sup.[cross product] (n+1)]) for n [greater than or equal to] 1 follows from using the multiplication epi, split by the unit (e.g., see commutative diagram [41, (3.10)]), or the counit splitting the comultiplication monomorphism.

In fact by writing down the obvious commutative diagrams one sees that any two 2-extensions in m that are related (1.1) lead to the same [Phi](m).

Given an abelian category A assume that we have a commutative diagram in A with exact lines:

Smith, "Homotopy commutative diagrams and their realizations", J.

Thus the strict diagram (2.13) fits into a homotopy commutative diagram:

For any map [[iota].sub.X]: A [right arrow] X of T, the Ganea construction of [[iota].sub.X] is the following sequence of homotopy

commutative diagrams (i [greater than or equal to] 0):

(d) This follows from the following commutative diagram:

(2) For this, just recall that the bordism Kunneth formula is induced by the cofibration sequence associated with the canonical nontrivial map P [right arrow] C[P.sup.[infinity]], which accompanies the homotopy commutative diagram: