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International Journal of Mathematics and Mathematical Sciences
Volume 2005, Issue 1, Pages 155-162

Further remarks on systems of interlocking exact sequences

1Department of Mathematical Sciences, State University of New York at Binghamton, Binghamton 13902-6000, NY, USA
2Department of Mathematics and Computer Science, Providence College, Providence 02918, RI, USA

Received 10 May 2004

Copyright © 2005 Peter Hilton and C. Joanna Su. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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 1967, 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 noncrossing points. As to a crossing point p, we need the extra criterion that the fourth sequence is differential. 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.