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A Torsion‐Free Milnor–Moore Theorem
Author(s) -
Scott Jonathan A.
Publication year - 2003
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/s002461070300423x
Subject(s) - subring , mathematics , invertible matrix , homomorphism , torsion (gastropod) , embedding , isomorphism (crystallography) , combinatorics , pure mathematics , discrete mathematics , crystallography , ring (chemistry) , crystal structure , medicine , chemistry , surgery , organic chemistry , artificial intelligence , computer science
Let Ω X be the space of Moore loops on a finite, q ‐connected, n ‐dimensional CW complex X , and let R ⊂ Q be a subring containing 1/2. Let ρ( R ) be the least non‐invertible prime in R . For a graded R ‐module M of finite type, let FM = M /Torsion M . We show that the inclusion P ⊂ F H * (Ω X ; R ) of the sub‐Lie algebra of primitive elements induces an isomorphism of Hopf algebrasU P → ≅ F H * ( Ω X ; R ) ,provided that ρ( R ) ⩾ n/q . Furthermore, the Hurewicz homomorphism induces an embedding of F (π * (Ω X )⊗ R ) in P , with P/F (π * (Ω X )⊗ R ) torsion. As a corollary, if X is elliptic, then FH * (Ω X ; R ) is a finitely generated R ‐algebra.