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The fundamental group of a Peano continuum The first shape homomorphism The quasitopological fundamental group Comparing the approaches The quasitopological fundamental group and the first shape map Jeremy Brazas 28th Summer Conference on Topology and Its Applications North Bay, Ontario, Canada July 26, 2013 Jeremy Brazas The quasitopological fundamental group and the first shape map

The fundamental group of a Peano continuum The first shape homomorphism The quasitopological fundamental group Comparing the approaches Introduction Joint with Paul Fabel. ◮ J. Brazas, P . Fabel, Thick Spanier groups and the first shape map, To appear in Rocky Mountain J. Math. ◮ J. Brazas, P . Fabel, On fundamental groups with the quotient topology, To appear in J. Homotopy and Related Structures. 2013. Jeremy Brazas The quasitopological fundamental group and the first shape map

The fundamental group of a Peano continuum The first shape homomorphism The quasitopological fundamental group Comparing the approaches Introduction Joint with Paul Fabel. ◮ J. Brazas, P . Fabel, Thick Spanier groups and the first shape map, To appear in Rocky Mountain J. Math. ◮ J. Brazas, P . Fabel, On fundamental groups with the quotient topology, To appear in J. Homotopy and Related Structures. 2013. Jeremy Brazas The quasitopological fundamental group and the first shape map

The fundamental group of a Peano continuum The first shape homomorphism The quasitopological fundamental group Comparing the approaches The fundamental group The fundamental group π 1 ( X , x 0 ) of a Peano continuum X , x 0 ∈ X is either finitely presented (when X has a universal covering) ◮ or uncountable (when X does not have a universal covering) Jeremy Brazas The quasitopological fundamental group and the first shape map

The fundamental group of a Peano continuum The first shape homomorphism The quasitopological fundamental group Comparing the approaches The fundamental group The fundamental group π 1 ( X , x 0 ) of a Peano continuum X , x 0 ∈ X is either finitely presented (when X has a universal covering) ◮ or uncountable (when X does not have a universal covering) Motivation/Application: ◮ Distinguish homotopy types Jeremy Brazas The quasitopological fundamental group and the first shape map

The fundamental group of a Peano continuum The first shape homomorphism The quasitopological fundamental group Comparing the approaches The fundamental group The fundamental group π 1 ( X , x 0 ) of a Peano continuum X , x 0 ∈ X is either finitely presented (when X has a universal covering) ◮ or uncountable (when X does not have a universal covering) Motivation/Application: ◮ Distinguish homotopy types ◮ Provides new direction for combinatorial theory of infinitely generated groups, i.e. slender/n-slender/n-cotorsion free groups (Eda, Fischer) Jeremy Brazas The quasitopological fundamental group and the first shape map

The fundamental group of a Peano continuum The first shape homomorphism The quasitopological fundamental group Comparing the approaches The fundamental group The fundamental group π 1 ( X , x 0 ) of a Peano continuum X , x 0 ∈ X is either finitely presented (when X has a universal covering) ◮ or uncountable (when X does not have a universal covering) Motivation/Application: ◮ Distinguish homotopy types ◮ Provides new direction for combinatorial theory of infinitely generated groups, i.e. slender/n-slender/n-cotorsion free groups (Eda, Fischer) ◮ Natural topologies on homotopical invariants provide (wild) geometric models for objects in topological algebra. Jeremy Brazas The quasitopological fundamental group and the first shape map

The fundamental group of a Peano continuum The first shape homomorphism The quasitopological fundamental group Comparing the approaches The Hawaiian earring H Jeremy Brazas The quasitopological fundamental group and the first shape map

The fundamental group of a Peano continuum The first shape homomorphism The quasitopological fundamental group Comparing the approaches The Hawaiian earring H �� n � i = 1 S 1 , 0 The homomorphisms π 1 ( H , 0 ) → π 1 = F ( x 1 , ..., x n ) induce a canonical homomorphism Ψ : π 1 ( H , 0 ) → lim F ( x 1 , ..., x n ) ← − n Jeremy Brazas The quasitopological fundamental group and the first shape map

The fundamental group of a Peano continuum The first shape homomorphism The quasitopological fundamental group Comparing the approaches The Hawaiian earring H �� n � i = 1 S 1 , 0 The homomorphisms π 1 ( H , 0 ) → π 1 = F ( x 1 , ..., x n ) induce a canonical homomorphism Ψ : π 1 ( H , 0 ) → lim F ( x 1 , ..., x n ) ← − n Theorem (Griffiths, Morgan, Morrison): ker Ψ = 1 so Ψ is injective. Jeremy Brazas The quasitopological fundamental group and the first shape map

The fundamental group of a Peano continuum The first shape homomorphism The quasitopological fundamental group Comparing the approaches The Hawaiian earring H �� n � i = 1 S 1 , 0 The homomorphisms π 1 ( H , 0 ) → π 1 = F ( x 1 , ..., x n ) induce a canonical homomorphism Ψ : π 1 ( H , 0 ) → lim F ( x 1 , ..., x n ) ← − n Theorem (Griffiths, Morgan, Morrison): ker Ψ = 1 so Ψ is injective. An element in π 1 ( H , 0 ) = Im (Ψ) is a sequence ( w 1 , w 2 , ... ) where w n ∈ F ( x 1 , ..., x n ) and for every fixed generator x i the number of times x i appears in w n is eventually constant. Jeremy Brazas The quasitopological fundamental group and the first shape map

The fundamental group of a Peano continuum The first shape homomorphism The quasitopological fundamental group Comparing the approaches The ˇ Cech expansion Choose a finite open cover U n of X consisting of path connected open balls U with diam ( U ) < 1 n such that U n + 1 � U n (refinement). Jeremy Brazas The quasitopological fundamental group and the first shape map

The fundamental group of a Peano continuum The first shape homomorphism The quasitopological fundamental group Comparing the approaches The ˇ Cech expansion Choose a finite open cover U n of X consisting of path connected open balls U with diam ( U ) < 1 n such that U n + 1 � U n (refinement). Let X n = N ( U n ) be the nerve of U n . Jeremy Brazas The quasitopological fundamental group and the first shape map

The fundamental group of a Peano continuum The first shape homomorphism The quasitopological fundamental group Comparing the approaches The ˇ Cech expansion Choose a finite open cover U n of X consisting of path connected open balls U with diam ( U ) < 1 n such that U n + 1 � U n (refinement). Let X n = N ( U n ) be the nerve of U n . Refinement gives an inverse sequence of polyhedra � X n + 1 p n + 1 , n � X n � X 2 p 2 , 1 � X 1 p n , n − 1 � · · · · · · Jeremy Brazas The quasitopological fundamental group and the first shape map

The fundamental group of a Peano continuum The first shape homomorphism The quasitopological fundamental group Comparing the approaches The fundamental pro-group The fundamental pro-group is the inverse sequence ( π 1 ( X n , x n ) , ( p n + 1 , n ) ∗ ) of finitely generated groups. Jeremy Brazas The quasitopological fundamental group and the first shape map

The fundamental group of a Peano continuum The first shape homomorphism The quasitopological fundamental group Comparing the approaches The fundamental pro-group The fundamental pro-group is the inverse sequence ( π 1 ( X n , x n ) , ( p n + 1 , n ) ∗ ) of finitely generated groups. The first shape homotopy group is ˇ π 1 ( X , x 0 ) = lim − ( π 1 ( X n , x n ) , ( p n + 1 , n ) ∗ ) . ← Jeremy Brazas The quasitopological fundamental group and the first shape map

The fundamental group of a Peano continuum The first shape homomorphism The quasitopological fundamental group Comparing the approaches The fundamental pro-group The fundamental pro-group is the inverse sequence ( π 1 ( X n , x n ) , ( p n + 1 , n ) ∗ ) of finitely generated groups. The first shape homotopy group is ˇ π 1 ( X , x 0 ) = lim − ( π 1 ( X n , x n ) , ( p n + 1 , n ) ∗ ) . ← Using partions of unity, construct canonical maps p n : X → X n such that p n + 1 , n ◦ p n + 1 ≃ p n Jeremy Brazas The quasitopological fundamental group and the first shape map

� � � The fundamental group of a Peano continuum The first shape homomorphism The quasitopological fundamental group Comparing the approaches The fundamental pro-group The fundamental pro-group is the inverse sequence ( π 1 ( X n , x n ) , ( p n + 1 , n ) ∗ ) of finitely generated groups. The first shape homotopy group is ˇ π 1 ( X , x 0 ) = lim − ( π 1 ( X n , x n ) , ( p n + 1 , n ) ∗ ) . ← Using partions of unity, construct canonical maps p n : X → X n such that p n + 1 , n ◦ p n + 1 ≃ p n π 1 ( X , x 0 ) ( pn ) ∗ ( p 1 ) ∗ ( p 2 ) ∗ ( pn + 1 , n ) ∗ � π 1 ( X n , x n ) � · · · � π 1 ( X 2 , x 2 ) ( p 2 , 1 ) ∗ � π 1 ( X 1 , x 1 ) · · · Jeremy Brazas The quasitopological fundamental group and the first shape map