Tree-adjoined spaces and the Hawaiian earring

Transcription

1 Tree-adjoined spaces and the Hawaiian earring W. Hojka (TU Wien) Workshop on Fractals and Tilings 2009 July 6-10, 2009, Strobl (Austria) W. Hojka (TU Wien) () Tree-adjoined spaces and the Hawaiian earring Fractals and Tilings / 7

2 Wild topology In many interesting cases of wild spaces, their fundamental group can be embedded in an inverse limit of free groups. Elements in that limit can be considered as infinite words, e.g. for the Hawaiian earring: W. Hojka (TU Wien) () Tree-adjoined spaces and the Hawaiian earring Fractals and Tilings / 7

3 Wild topology In many interesting cases of wild spaces, their fundamental group can be embedded in an inverse limit of free groups. Elements in that limit can be considered as infinite words, e.g. for the Hawaiian earring: Definition Let A = {a 1, a 2,...}, and W be the set of all the maps g : λ A A 1, where λ is a countable linear order type and the preimage of each a i is finite. Let W be the quotient set, where two elements are identified, whenever there are cancellations that reduce them to the same word. W. Hojka (TU Wien) () Tree-adjoined spaces and the Hawaiian earring Fractals and Tilings / 7

4 Wild topology In many interesting cases of wild spaces, their fundamental group can be embedded in an inverse limit of free groups. Elements in that limit can be considered as infinite words, e.g. for the Hawaiian earring: Definition Let A = {a 1, a 2,...}, and W be the set of all the maps g : λ A A 1, where λ is a countable linear order type and the preimage of each a i is finite. Let W be the quotient set, where two elements are identified, whenever there are cancellations that reduce them to the same word. Theorem Together with concatenation of orders, W forms a group and is isomorphic to the fundamental group π 1 (H, x 0 ) of the Hawaiian earring. W. Hojka (TU Wien) () Tree-adjoined spaces and the Hawaiian earring Fractals and Tilings / 7

5 Making spaces wild W. Hojka (TU Wien) () Tree-adjoined spaces and the Hawaiian earring Fractals and Tilings / 7

6 Making spaces wild Let Y be a compact metric space. Theorem There is a surjective map f from the Cantor set C onto Y. W. Hojka (TU Wien) () Tree-adjoined spaces and the Hawaiian earring Fractals and Tilings / 7

7 Making spaces wild Let Y be a compact metric space. Theorem There is a surjective map f from the Cantor set C onto Y. Theorem Consider C as a subset of the unit interval I. The adjunction space X f := Y f I is a compact metric space, that is wild at every point in Y. Alternatively, instead of I, an infinite binary tree attached to the cantor set can be used. Both construction yield homotopy equivalent spaces. W. Hojka (TU Wien) () Tree-adjoined spaces and the Hawaiian earring Fractals and Tilings / 7

8 A sequence of maps Y is embedded canonically into X f. By contracting Y to a single point, there is a natural map from X f onto H. W. Hojka (TU Wien) () Tree-adjoined spaces and the Hawaiian earring Fractals and Tilings / 7

9 A sequence of maps Y is embedded canonically into X f. By contracting Y to a single point, there is a natural map from X f onto H. Both maps induce homomorphisms on the corresponding fundamental groups: Some properties (i) ι is injective, π 1 Y ι π 1 X ψ π 1 H W. Hojka (TU Wien) () Tree-adjoined spaces and the Hawaiian earring Fractals and Tilings / 7

10 A sequence of maps Y is embedded canonically into X f. By contracting Y to a single point, there is a natural map from X f onto H. Both maps induce homomorphisms on the corresponding fundamental groups: Some properties (i) ι is injective, (ii) im ι ker ψ, π 1 Y ι π 1 X ψ π 1 H W. Hojka (TU Wien) () Tree-adjoined spaces and the Hawaiian earring Fractals and Tilings / 7

11 A sequence of maps Y is embedded canonically into X f. By contracting Y to a single point, there is a natural map from X f onto H. Both maps induce homomorphisms on the corresponding fundamental groups: Some properties (i) ι is injective, (ii) im ι ker ψ, π 1 Y ι π 1 X ψ π 1 H (iii) If Y is connected and locally simply connected, then (im ι) π 1X = ker ψ. W. Hojka (TU Wien) () Tree-adjoined spaces and the Hawaiian earring Fractals and Tilings / 7

12 Sketch of proof of (iii) Lemma If a small path γ is trivial, then there is a small homotopy between γ and 1: W. Hojka (TU Wien) () Tree-adjoined spaces and the Hawaiian earring Fractals and Tilings / 7

13 Sketch of proof of (iii) Lemma If a small path γ is trivial, then there is a small homotopy between γ and 1: ε δ γ 1 : D(im γ) < δ H homotopy between γ and 1, and D(im H) < ε (where D is the diameter of a set). W. Hojka (TU Wien) () Tree-adjoined spaces and the Hawaiian earring Fractals and Tilings / 7

14 Sketch of proof of (iii) Lemma If a small path γ is trivial, then there is a small homotopy between γ and 1: ε δ γ 1 : D(im γ) < δ H homotopy between γ and 1, and D(im H) < ε (where D is the diameter of a set). Let γ : I X f be a path that is in the kernel of ψ. Find a cancellation on γ 1 (X f \ Y ). That induces a relation on a partition on the rest of I. W. Hojka (TU Wien) () Tree-adjoined spaces and the Hawaiian earring Fractals and Tilings / 7

15 Sketch of proof of (iii) Lemma If a small path γ is trivial, then there is a small homotopy between γ and 1: ε δ γ 1 : D(im γ) < δ H homotopy between γ and 1, and D(im H) < ε (where D is the diameter of a set). Let γ : I X f be a path that is in the kernel of ψ. Find a cancellation on γ 1 (X f \ Y ). That induces a relation on a partition on the rest of I. Only finitely many induced sub-paths in Y are not nullhomotopic. W. Hojka (TU Wien) () Tree-adjoined spaces and the Hawaiian earring Fractals and Tilings / 7

16 Sketch of proof of (iii) Lemma If a small path γ is trivial, then there is a small homotopy between γ and 1: ε δ γ 1 : D(im γ) < δ H homotopy between γ and 1, and D(im H) < ε (where D is the diameter of a set). Let γ : I X f be a path that is in the kernel of ψ. Find a cancellation on γ 1 (X f \ Y ). That induces a relation on a partition on the rest of I. Only finitely many induced sub-paths in Y are not nullhomotopic. Remove convex nullhomotopic sub-intervals by a common homotopy. W. Hojka (TU Wien) () Tree-adjoined spaces and the Hawaiian earring Fractals and Tilings / 7

17 Sketch of proof of (iii), continued Insert (finitely many) return paths, connecting the subpath to the base point. W. Hojka (TU Wien) () Tree-adjoined spaces and the Hawaiian earring Fractals and Tilings / 7

18 Sketch of proof of (iii), continued Insert (finitely many) return paths, connecting the subpath to the base point. This yields a homotopy between the path and a finite concatenation of loops in ΩX f, γ w 1 w n. W. Hojka (TU Wien) () Tree-adjoined spaces and the Hawaiian earring Fractals and Tilings / 7

19 Sketch of proof of (iii), continued Insert (finitely many) return paths, connecting the subpath to the base point. This yields a homotopy between the path and a finite concatenation of loops in ΩX f, γ w 1 w n. Induction shows that this word corresponds to an element in (im ι) π 1X f, the normal subgroup generated by the kernel. W. Hojka (TU Wien) () Tree-adjoined spaces and the Hawaiian earring Fractals and Tilings / 7

20 Sketch of proof of (iii), continued Insert (finitely many) return paths, connecting the subpath to the base point. This yields a homotopy between the path and a finite concatenation of loops in ΩX f, γ w 1 w n. Induction shows that this word corresponds to an element in (im ι) π 1X f, the normal subgroup generated by the kernel. Thus, ker ψ = (im ι) π 1X f. W. Hojka (TU Wien) () Tree-adjoined spaces and the Hawaiian earring Fractals and Tilings / 7

21 Sketch of proof of (iii), continued Insert (finitely many) return paths, connecting the subpath to the base point. This yields a homotopy between the path and a finite concatenation of loops in ΩX f, γ w 1 w n. Induction shows that this word corresponds to an element in (im ι) π 1X f, the normal subgroup generated by the kernel. Thus, ker ψ = (im ι) π 1X f. Corollary If Y is simply connected, the fundamental group of X f injects into that of the Hawaiian earring. W. Hojka (TU Wien) () Tree-adjoined spaces and the Hawaiian earring Fractals and Tilings / 7

22 A group induced relation Take any subset S π 1 H < lim F n. W. Hojka (TU Wien) () Tree-adjoined spaces and the Hawaiian earring Fractals and Tilings / 7

23 A group induced relation Take any subset S π 1 H < lim F n. Definition Define a relation on the Cantor set w.r.t. S: x y : w S k λ w : k order isomorphic to ω or ω, x, y cl A C (im w k). W. Hojka (TU Wien) () Tree-adjoined spaces and the Hawaiian earring Fractals and Tilings / 7

24 A group induced relation Take any subset S π 1 H < lim F n. Definition Define a relation on the Cantor set w.r.t. S: x y : w S k λ w : k order isomorphic to ω or ω, x, y cl A C (im w k). Theorem S = im ψ C/ = Y. W. Hojka (TU Wien) () Tree-adjoined spaces and the Hawaiian earring Fractals and Tilings / 7

25 A group induced relation Take any subset S π 1 H < lim F n. Definition Define a relation on the Cantor set w.r.t. S: x y : w S k λ w : k order isomorphic to ω or ω, x, y cl A C (im w k). Theorem S = im ψ C/ = Y. Remark f 1, f 2 : C Y X f1 h.e. X f2, σ permutation on the alphabet, (im ψ 1 ) σ = im ψ 2. W. Hojka (TU Wien) () Tree-adjoined spaces and the Hawaiian earring Fractals and Tilings / 7