The Genus of a Map
Copyright policy )The Genus of a Map
The elements [ f ′ ] ( f ′ : X ′ → Y ′ ) [f’](f’:X’ \to Y’) of the genus − G ( f ) - G(f) of a map f : X → Y f:X \to Y are equivalence classes of homotopy classes of maps f ′ f’ which satisfy: For every prime p p there exist homotopy equivalences h p : X p ′ → X p {h_p}:{X’_p} \to {X_p} and k p : Y p ′ → Y p {k_p}:{Y’_p} \to {Y_p} so that f p h p ∼ k p f p ′ {f_p}{h_p} \sim {k_p}{f’_p} . The genus of f f under X − G X ( f ) X - {G^X}(f) and the genus of f f over Y − G Y ( f ) Y - {G_Y}(f) are defined similarly. In this paper we prove that under certain conditions on f f , the sets G ( f ) G(f) , G X ( f ) {G^X}(f) and G Y ( f ) {G_Y}(f) are finite and admit an abelian group structure. We also compare the genus of f f to those of X X and Y Y , calculate it for some principal fibrations of the form K ( G , n − 1 ) → X → Y K(G,n - 1) \to X \to Y , and deal with the noncancellation phenomenon.
homotopy equivalences at a prime, Fiber spaces in algebraic topology, Localization and completion in homotopy theory, Homotopy groups, general; sets of homotopy classes, genus of a map, principal fibrations with an Eilenberg-MacLane space as fibre, Homotopy equivalences in algebraic topology, Loop spaces
homotopy equivalences at a prime, Fiber spaces in algebraic topology, Localization and completion in homotopy theory, Homotopy groups, general; sets of homotopy classes, genus of a map, principal fibrations with an Eilenberg-MacLane space as fibre, Homotopy equivalences in algebraic topology, Loop spaces
selected citations These citations are derived from selected sources.This is an alternative to the "Influence" indicator, which also reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).0 popularity This indicator reflects the "current" impact/attention (the "hype") of an article in the research community at large, based on the underlying citation network.Average influence This indicator reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).Average impulse This indicator reflects the initial momentum of an article directly after its publication, based on the underlying citation network.Average
Citations provided by BIP!Popularity provided by BIP!