Algebraic Topology – A Student's Guide

Adams, J. F.; Shepherd, G. C.

doi:10.1017/cbo9780511662584

Cited by 47 publications

(17 citation statements)

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“…, a G A. We note that (/^""'/(^^(Iir)*" 1 ®.. Z and that, if H X X is finitelygenerated, so is TTIX and hence so is (/TT) 1 '" 1 as ir-module. Thus on either hypothesis on the class C, we may infer that, <f> 2 being C-surjective, so is l®</>2 and hence also </>j!+i, fc^2.…”

Section: Nilpotent Local Coefficientsmentioning

confidence: 95%

“…We shall, as indicated in the preamble, restrict ourselves to commenting on the various places in the proof given in [1] where modification is necessary.…”

Section: The Comparison Theorem For Maps Of Fibre Spacesmentioning

confidence: 99%

“…Finally, in Step 3, ( [1]; p. Ill, 112), we use Corollary 3.4 to infer from the assertion H r (A 2 , C 2 )eS, r<p + q-2, that also i7 r (A 2 , C 2 )eS, r< p + q -2. In fact, the hypotheses p2=3, q>3, needed elsewhere in the proof, imply a fortiori that p + q-2>2, thus insuring fulfillment of the dimensionality criterion in Corollary 3.4.…”

Section: The Comparison Theorem For Maps Of Fibre Spacesmentioning

confidence: 99%

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On the Zeeman Comparison Theorem for the Homology of Quasi-Nilpotent Fibrations

Hilton

Roitberg

1976

Q J Math

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THIS note constitutes a generalization of the Zeeman comparison theorem for spectral sequences [9]. Zeeman's theorem was based on hypotheses valid for the homology spectral sequence of a fibration with simplyconnected base space; however, his hypotheses were stated purely algebraically and there was no assumption that the spectral sequence was derived from a filtered chain complex-merely that it was a first quadrant sequence and the differentials had the usual bidegrees. Thus Zeeman's version was more general than Moore's earlier comparison theorem [5]; it was also more general in that isomorphism assumptions were only made up to certain dimensions (and so only deduced up to certain dimensions).We generalize Zeeman's theorem in two directions. The most important direction is that we cover the situation of a quasi-nilpotent fibration; this is a fibrationin which all spaces are connected, and TTIB operates nilpotently on H t F, i"3=0. We say that (0.1) is strongly quasi-nilpotent if it is quasi-nilpotent and if, in addition, inB is nilpotent. Among the quasi-nilpotent fibrations we find the nilpotent fibrations [2]; these are fibrations (0.1) in which all spaces are connected and 7T]E operates nilpotently [3] on 7T ( F, i5»l. If E, B are nilpotent spaces and F is connected then F is nilpotent and (0.1) is a nilpotent fibration, and also strongly quasi-nilpotent. A special case, then, of a nilpotent fibration which is also strongly quasi-nilpotent iswhere X is nilpotent and X is the universal cover of X. We obtain a very general theorem of Whitehead type by applying our comparison theorem to the homology spectral sequence of (0.2) (see Corollary 3.4); we remark that this spectral sequence violates not only assumption (iii) of [9] but also the refinement of (iii) mentioned on p. 58 of [9]. We should mention at this point that our results strengthen those of Zeeman even when the base is simply-connected-that is, in the case Quart.

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Section: Nilpotent Local Coefficientsmentioning

confidence: 95%

“…We shall, as indicated in the preamble, restrict ourselves to commenting on the various places in the proof given in [1] where modification is necessary.…”

Section: The Comparison Theorem For Maps Of Fibre Spacesmentioning

confidence: 99%

Section: The Comparison Theorem For Maps Of Fibre Spacesmentioning

confidence: 99%

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On the Zeeman Comparison Theorem for the Homology of Quasi-Nilpotent Fibrations

Hilton

Roitberg

1976

Q J Math

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“…In other words, X is a path-connected pointed space with two path-connected subspaces A and B, containing the basepoint, such that the intersection W = A ∩ B is path connected. Consider the triad homotopy group π 3 (X; A, B); see for example [1,6]. There are obvious morphisms π 3 (X; A, B) → π 2 (A, W ) and π 3 (X; A, B) → π 2 (B, W ).…”

Section: The Crossed Pairingmentioning

confidence: 99%

“…There also exists an action of π 1 (W ) on all of these groups, and all boundary maps are π 1 (W )-equivariant. The generalised Whitehead product [1,5] …”

Section: The Crossed Pairingmentioning

confidence: 99%

The fundamental 2-crossed complex of a reduced CW-complex

Martins

2011

Homology, Homotopy and Applications

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We define the fundamental 2-crossed complex Ω ∞ (X) of a reduced CW-complex X from Ellis' fundamental squared complex ρ ∞ (X) thereby proving that Ω ∞ (X) is totally free on the set of cells of X. This fundamental 2-crossed complex has very good properties with regard to the geometrical realisation of 2-crossed complex morphisms. After carefully discussing the homotopy theory of totally free 2-crossed complexes, we use Ω ∞ (X) to give a new proof that the homotopy category of pointed 3-types is equivalent to the homotopy category of 2-crossed modules of groups. We obtain very similar results to the ones given by Baues in the similar context of quadratic modules and quadratic chain complexes.

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