On the homology theory of central group extensions II. The exact sequence in the general case

1972 ◽  
Vol 47 (1) ◽  
pp. 171-178 ◽  
Author(s):  
Beno Eckmann ◽  
Peter Hilton ◽  
Urs Stammbach
Keyword(s):  
Exact Sequence ◽  
Homology Theory ◽  
Central Group ◽  
Group Extensions

On the homology theory of central group extensions: I—The commutator map and stem extensions

1972 ◽  
Vol 47 (1) ◽  
pp. 102-122 ◽  
Author(s):  
Beno Eckmann ◽  
Peter J. Hilton ◽  
Urs Stammbach
Keyword(s):  
Homology Theory ◽  
Central Group ◽  
Group Extensions

scholarly journals On the homology of central group extensions II. The exact sequence in the general case

1973 ◽  
Vol 48 (1) ◽  
pp. 136-136
Author(s):  
Beno Eckmann ◽  
Peter Hilton ◽  
Urs Stammbach
Keyword(s):  
Exact Sequence ◽  
Central Group ◽  
Group Extensions

scholarly journals A note on central group extensions

1973 ◽  
Vol 15 (4) ◽  
pp. 428-429 ◽  
Author(s):  
G. J. Hauptfleisch
Keyword(s):  
Abelian Group ◽  
Homomorphic Image ◽  
Free Group ◽  
Central Extension ◽  
Dihedral Group ◽  
Group Extension ◽  
Central Group ◽  
Group Extensions

If A, B, H, K are abelian group and φ: A → H and ψ: B → K are epimorphisms, then a given central group extension G of H by K is not necessarily a homomorphic image of a group extension of A by B. Take for instance A = Z(2), B = Z ⊕ Z, H = Z(2), K = V4 (Klein's fourgroup). Then the dihedral group D8 is a central extension of H by K but it is not a homomorphic image of Z ⊕ Z ⊕ Z(2), the only group extension of A by the free group B.

On the relation of connective K-theory to homology

1970 ◽  
Vol 68 (3) ◽  
pp. 637-639 ◽  
Author(s):  
Larry Smith
Keyword(s):  
Exact Sequence ◽  
Natural Transformation ◽  
Homology Theory ◽  
Finite Complex ◽  
Ring Spectrum ◽  
Bott Periodicity ◽  
Ring Spectra ◽  
K Theory ◽  
Image Position ◽  
Natural Way

Let us denote by k*( ) the homology theory determined by the connective BU spectrum, bu, that is, in the notations of (1) and (9), bu2n = BU(2n,…,∞), bu2n+1 = U(2n + 1,…, ∞) with the spectral maps induced via Bott periodicity. The resulting spectrum, bu, is a ring spectrum. Recall that k*(point) ≅ Z[t], degree t = 2. There is a natural transformation of ring spectrainducing a morphismof homology functors. It is the objective of this note to establish: Theorem. Let X be a finite complex. Then there is a natural exact sequencewhere Z is viewed as a Z[t] module via the augmentationand, is induced by η*in the natural way.

scholarly journals Source Algebras of p-Central Group Extensions

2001 ◽  
Vol 235 (1) ◽  
pp. 359-398 ◽  
Author(s):  
Lluis Puig
Keyword(s):  
Central Group ◽  
Group Extensions

scholarly journals GEOMETRIC INTERPRETATIONS OF QUANDLE HOMOLOGY

2001 ◽  
Vol 10 (03) ◽  
pp. 345-386 ◽  
Author(s):  
J. SCOTT CARTER ◽  
SEIICHI KAMADA ◽  
MASAHICO SAITO
Keyword(s):  
Exact Sequence ◽  
Homology Theory ◽  
Exceptional Points ◽  
Geometric Representations ◽  
Knot Diagrams

Geometric representations of cycles in quandle homology theory are given in terms of colored knot diagrams. Abstract knot diagrams are generalized to diagrams with exceptional points which, when colored, correspond to degenerate cycles. Bounding chains are realized, and used to obtain equivalence moves for homologous cycles. The methods are applied to prove that boundary homomorphisms in a homology exact sequence vanish.

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