Contributions to certain problems in algebraic topology

• We consider two problems. The first one is to determine when a fibration is also a (homotopy) cofibration. We use a classic result of T. Ganea to give a complete algebraic solution. When the correspondence between topology and algebra is very close (i.e., when the total space of the fibration is nilpotent} we show that such fibrations can be described quite explicitly and that, in the absence of nilpotency, no such simple characterization exists.The second problem is to find a topological interpretation of the property of having finite quasi-projective dimension (qpd). Groups with this property were introduced by J. Howie and H.R. Schneebeli as a generalization of the Identity Property. We begin by showing that if the group G has qpd G = n, then the relative cohomological dimension of G, rd G, (relative to the maximal finite subgroups of G) is at most n, and we make precise the relationship between qpd G and rd G.By introducing 0-unfree actions we obtain a topological interpretation of the property of having finite rd. We then give the desired interpretation of finite qpd in terms of 0-unfree, periodic actions.Finally, we extend a related result of C.T.C. Wall. Let IG denote the augmentation ideal of G and, if H is a subgroup of G, let JH(G) denote the ideal of ZZG generated by IH. Wall has shown (as sharpened by Dunwoody) that if JH(G) is a direct summand of IG, the pair (G,H) is accessible and Δ= IG/ JH(G)  is a ZZG -projective module, then H is a free factor of G. We relax the last condition by demanding instead that the projective dimension of Δ over ZZG be finite.

Ämnesord

NATURVETENSKAP  -- Matematik (hsv//swe)

NATURAL SCIENCES  -- Mathematics (hsv//eng)

Nyckelord

Algebraisk topologi

Publikations- och innehållstyp

vet (ämneskategori)

dok (ämneskategori)