Introduction Results Proofs Cotorsion-free groups from a topological viewpoint Hanspeter Fischer (Ball State University, USA) joint work with Katsuya Eda (Waseda University, Japan) TOPOSYM 2016 Prague, Czech Republic July 26, 2016 K. Eda, H. Fischer Cotorsion-free groups from a topological viewpoint
Introduction Spanier groups Results Generalized covering spaces Proofs Generalized slender groups Definition For an open cover U of a path-connected space X and x ∈ X , π (U , x ) ⩽ π 1 ( X , x ) is generated by all elements [ αβα − ] with α ∶ ([ 0 , 1 ] , 0 ) → ( X , x ) , β ∶ ([ 0 , 1 ] , { 0 , 1 }) → ( U ,α ( 1 )) , U ∈ U . x α U ∈ U β Generalized covering spaces Generalized slender groups Asphericity criteria Theory of free σ -products “Cayley graph” for π 1 ( M ) Classification of homotopy of the Menger curve M types of 1-dim spaces by π 1 ⋮ ⋮ K. Eda, H. Fischer Cotorsion-free groups from a topological viewpoint
Introduction Spanier groups Results Generalized covering spaces Proofs Generalized slender groups Properties (1) π ( U , x ) is a normal subgroup of π 1 ( X , x ) . K. Eda, H. Fischer Cotorsion-free groups from a topological viewpoint
Introduction Spanier groups Results Generalized covering spaces Proofs Generalized slender groups Properties (1) π ( U , x ) is a normal subgroup of π 1 ( X , x ) . (2) If V refines U , then π ( V , x ) ⩽ π ( U , x ) . K. Eda, H. Fischer Cotorsion-free groups from a topological viewpoint
Introduction Spanier groups Results Generalized covering spaces Proofs Generalized slender groups Properties (1) π ( U , x ) is a normal subgroup of π 1 ( X , x ) . (2) If V refines U , then π ( V , x ) ⩽ π ( U , x ) . (3) For locally path-connected X : (a) ∃ V ∶ π ( V , x ) = ⋂ π ( U , x ) ⇔ X has a universal covering space. U K. Eda, H. Fischer Cotorsion-free groups from a topological viewpoint
Introduction Spanier groups Results Generalized covering spaces Proofs Generalized slender groups Properties (1) π ( U , x ) is a normal subgroup of π 1 ( X , x ) . (2) If V refines U , then π ( V , x ) ⩽ π ( U , x ) . (3) For locally path-connected X : (a) ∃ V ∶ π ( V , x ) = ⋂ π ( U , x ) ⇔ X has a universal covering space. U (b) ∃ U ∶ π ( U , x ) = 1 ⇔ X has a simply connected covering space. K. Eda, H. Fischer Cotorsion-free groups from a topological viewpoint
Introduction Spanier groups Results Generalized covering spaces Proofs Generalized slender groups Properties (1) π ( U , x ) is a normal subgroup of π 1 ( X , x ) . (2) If V refines U , then π ( V , x ) ⩽ π ( U , x ) . (3) For locally path-connected X : (a) ∃ V ∶ π ( V , x ) = ⋂ π ( U , x ) ⇔ X has a universal covering space. U (b) ∃ U ∶ π ( U , x ) = 1 ⇔ X has a simply connected covering space. Example: The Hawaiian Earring H = ∞ ⋃ C k k = 1 ∀ U ∶ π ( U , ∗ ) / = 1 C 1 but ⋂ π ( U , ∗ ) = 1 C 2 C 3 U K. Eda, H. Fischer Cotorsion-free groups from a topological viewpoint
Introduction Spanier groups Results Generalized covering spaces Proofs Generalized slender groups Properties (1) π ( U , x ) is a normal subgroup of π 1 ( X , x ) . (2) If V refines U , then π ( V , x ) ⩽ π ( U , x ) . (3) For locally path-connected X : (a) ∃ V ∶ π ( V , x ) = ⋂ π ( U , x ) ⇔ X has a universal covering space. U (b) ∃ U ∶ π ( U , x ) = 1 ⇔ X has a simply connected covering space. Example: The Hawaiian Earring H = ∞ ⋃ C k k = 1 ∀ U ∶ π ( U , ∗ ) / = 1 C 1 but ⋂ π ( U , ∗ ) = 1 C 2 C 3 U Definition: π s ( X , x ) = ⋂ π ( U , x ) ( Spanier group ) U K. Eda, H. Fischer Cotorsion-free groups from a topological viewpoint
� � Introduction Spanier groups Results Generalized covering spaces Proofs Generalized slender groups Theorem [F-Zastrow 2007] There exists a generalized covering p ∶ ̃ X → X w.r.t. π s ( X , x ) : (1) ̃ X is path connected (pc) and locally path connected (lpc). (2) p # ∶ π 1 ( ̃ X , ̃ x ) → π 1 ( X , x ) is a monomorphism onto π s ( X , x ) . ( ̃ X , ̃ x ) ⇒ f # π 1 ( Y , y ) ⩽ p # π 1 ( ̃ X , ̃ x ) ∃ ! ̃ f (3) ⇐ p ∀ f � ( X , x ) ( Y pc , lpc , y ) K. Eda, H. Fischer Cotorsion-free groups from a topological viewpoint
� � Introduction Spanier groups Results Generalized covering spaces Proofs Generalized slender groups Theorem [F-Zastrow 2007] There exists a generalized covering p ∶ ̃ X → X w.r.t. π s ( X , x ) : (1) ̃ X is path connected (pc) and locally path connected (lpc). (2) p # ∶ π 1 ( ̃ X , ̃ x ) → π 1 ( X , x ) is a monomorphism onto π s ( X , x ) . ( ̃ X , ̃ x ) ⇒ f # π 1 ( Y , y ) ⩽ p # π 1 ( ̃ X , ̃ x ) ∃ ! ̃ f (3) ⇐ p ∀ f � ( X , x ) ( Y pc , lpc , y ) If π s ( X , x ) = 1, we call p ∶ ̃ X → X a generalized universal covering . K. Eda, H. Fischer Cotorsion-free groups from a topological viewpoint
� � Introduction Spanier groups Results Generalized covering spaces Proofs Generalized slender groups Theorem [F-Zastrow 2007] There exists a generalized covering p ∶ ̃ X → X w.r.t. π s ( X , x ) : (1) ̃ X is path connected (pc) and locally path connected (lpc). (2) p # ∶ π 1 ( ̃ X , ̃ x ) → π 1 ( X , x ) is a monomorphism onto π s ( X , x ) . ( ̃ X , ̃ x ) ⇒ f # π 1 ( Y , y ) ⩽ p # π 1 ( ̃ X , ̃ x ) ∃ ! ̃ f (3) ⇐ p ∀ f � ( X , x ) ( Y pc , lpc , y ) If π s ( X , x ) = 1, we call p ∶ ̃ X → X a generalized universal covering . Examples with π s ( X , x ) = 1 include: 1-dimensional spaces [Eda-Kawamura 1998] subsets of surfaces [F-Zastrow 2005] certain “trees of manifolds” [F-Guilbault 2005] K. Eda, H. Fischer Cotorsion-free groups from a topological viewpoint
Introduction Spanier groups Results Generalized covering spaces Proofs Generalized slender groups Definition An abelian group A is called slender if for every homomorphism h ∶ Z N → A , ∃ n ∈ N ∀ c n , c n + 1 , ⋅⋅⋅ ∈ Z : h ( 0 ,..., 0 , c n , c n + 1 ,... ) = 0. K. Eda, H. Fischer Cotorsion-free groups from a topological viewpoint
Introduction Spanier groups Results Generalized covering spaces Proofs Generalized slender groups Definition An abelian group A is called slender if for every homomorphism h ∶ Z N → A , ∃ n ∈ N ∀ c n , c n + 1 , ⋅⋅⋅ ∈ Z : h ( 0 ,..., 0 , c n , c n + 1 ,... ) = 0. Definition A group G is called n-slender if for every homomorphism h ∶ π 1 ( H , ∗ ) → G , ∃ n ∈ N ∀ γ ⊆ ⋃ ∞ k = n C k : h ([ γ ]) = 1. K. Eda, H. Fischer Cotorsion-free groups from a topological viewpoint
Introduction Spanier groups Results Generalized covering spaces Proofs Generalized slender groups Definition An abelian group A is called slender if for every homomorphism h ∶ Z N → A , ∃ n ∈ N ∀ c n , c n + 1 , ⋅⋅⋅ ∈ Z : h ( 0 ,..., 0 , c n , c n + 1 ,... ) = 0. Definition A group G is called n-slender if for every homomorphism h ∶ π 1 ( H , ∗ ) → G , ∃ n ∈ N ∀ γ ⊆ ⋃ ∞ k = n C k : h ([ γ ]) = 1. Examples: Free groups are n-slender [Higman, Griffiths 1952-56]. Certain HNN extensions of n-slender groups are n-slender, including the Baumslag-Solitar groups [Nakamura 2015]. K. Eda, H. Fischer Cotorsion-free groups from a topological viewpoint
Introduction Spanier groups Results Generalized covering spaces Proofs Generalized slender groups Definition An abelian group A is called slender if for every homomorphism h ∶ Z N → A , ∃ n ∈ N ∀ c n , c n + 1 , ⋅⋅⋅ ∈ Z : h ( 0 ,..., 0 , c n , c n + 1 ,... ) = 0. Definition A group G is called n-slender if for every homomorphism h ∶ π 1 ( H , ∗ ) → G , ∃ n ∈ N ∀ γ ⊆ ⋃ ∞ k = n C k : h ([ γ ]) = 1. Examples: Free groups are n-slender [Higman, Griffiths 1952-56]. Certain HNN extensions of n-slender groups are n-slender, including the Baumslag-Solitar groups [Nakamura 2015]. Theorems [Eda 1992, 2005] (1) An abelian group A is n-slender ⇔ A is slender. K. Eda, H. Fischer Cotorsion-free groups from a topological viewpoint
Introduction Spanier groups Results Generalized covering spaces Proofs Generalized slender groups Definition An abelian group A is called slender if for every homomorphism h ∶ Z N → A , ∃ n ∈ N ∀ c n , c n + 1 , ⋅⋅⋅ ∈ Z : h ( 0 ,..., 0 , c n , c n + 1 ,... ) = 0. Definition A group G is called n-slender if for every homomorphism h ∶ π 1 ( H , ∗ ) → G , ∃ n ∈ N ∀ γ ⊆ ⋃ ∞ k = n C k : h ([ γ ]) = 1. Examples: Free groups are n-slender [Higman, Griffiths 1952-56]. Certain HNN extensions of n-slender groups are n-slender, including the Baumslag-Solitar groups [Nakamura 2015]. Theorems [Eda 1992, 2005] (1) An abelian group A is n-slender ⇔ A is slender. (2) A group G is n-slender ⇔ for every Peano continuum X and every homomorphism h ∶ π 1 ( X , x ) → G , ∃ U ∶ h ( π ( U , x )) = 1. K. Eda, H. Fischer Cotorsion-free groups from a topological viewpoint
Introduction Homomorphically Hausdorff Results Spanier-trivial Proofs Main Theorem Definition A group G is homomorphically Hausdorff relative to a space X if for every homomorphism h ∶ π 1 ( X , x ) → G , ⋂ h ( π (U , x )) = 1. U K. Eda, H. Fischer Cotorsion-free groups from a topological viewpoint
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