Further remarks on systems of interlocking exact sequences

In a system of interlocking sequences, the assumption that three out of the four sequences are exact does not guarantee the exactness of the fourth. In [1], Hilton proved that, with the additional condition that it is differential at the crossing points, the fourth sequence is also exact. In this paper, we trace such a diagram and analyze the relation between the kernels and the images, in the case that the fourth sequence is not necessarily exact. Regarding the exactness of the fourth sequence, we remark that the exactness of the other three sequences does guarantee the exactness of the fourth at non-crossing points. As to a crossing point p, we need the extra criterion that the fourth sequence is differential (see diagram (1.2)). One notices that the condition, for the fourth sequence, that kernel ⊇ image at p turns out to be equivalent to the ‘opposite’ condition kernel ⊆ image. Next, for the kernel and the image at p of the fourth sequence, even though they may not coincide, they are not far different – they always have the same cardinality as sets, and become isomorphic after taking quotients by a subgroup which is common to both. We demonstrate these phenomena with an example. 1 The Exactness of Systems of Interlocking Sequences It is well known that, in topology, the homotopy sequence of a triple (X,A,B), where X, A, and B are topological spaces with base points and B ⊆ A ⊆ X, is exact. In addition, the homotopy exact sequences of the pairs (A,B), (X,A), and (X,B), respectively, and the homotopy exact sequence of the triple (X,A,B) knot together in a one-of-a-kind commutative diagram, thus, · · · // πn+1(A,B) 1 // 4 && M M M M M M M M πn(B) 3 // 1 $$ I I I I I I I πn(X) 2 // 3 $$ J J J J J J J πn(X,A) // (( R R R · · · · · · 66 l l l · · · πn+1(X,B) 3 88 q q q q q q q q 4 && M M M M M M M M πn(A) 2 << x x x x x x x 1 "" F F F F F F F πn(X,B) 4 :: t t t t t t t 3 $$ J J J J J J J · · · (( R R R · · · · · · // πn+1(X) 2 // 3 88 q q q q q q q q πn+1(X,A) 4 // 2 :: u u u u u u u πn(A,B) 1 // 4 :: t t t t t t t πn−1(B) // 66 l l l · · · . (1.1) In [1], Hilton analyzed the general case of (1.1), namely, a commutative diagram that interlocks four sequences, of which three are known to be exact, and proved the following theorem: