Homotopy types of the letters of the alphabet

7. Main Idea: Nothing to it. Homotopy:

8. \(A \homotopic \Delta \homotopic S^1\)

9. \(a \homotopic d \homotopic o \homotopic S^1\)

10. \(B \homotopic 8 \homotopic S^1 \vee S^1\)

11. \(b \homotopic o \homotopic S^1\)

12. \(C \homotopic *\)

13. \(c \homotopic l \homotopic *\)

14. \(D \homotopic S^1\)

15. \(d \homotopic o \homotopic S^1\)

16. \(E \homotopic *\)

17. \(e \homotopic d \homotopic S^1\)

18. \(F \homotopic *\)

19. \(f \homotopic *\)

20. \(G \homotopic *\)

21. \(g \homotopic 8 \homotopic S^1 \vee S^1\)

22. \(H \homotopic *\)

23. \(h \homotopic l \homotopic *\)

24. \(I \homotopic *\)

25. \(i \homotopic \theset{*_1, *_2}\)

26. \(J \homotopic *\)

27. \(j \homotopic i \homotopic \theset{*_1, *_2}\)

28. \(K \homotopic *\)

    1. \(k \homotopic K \homotopic *\)

29. \(L \homotopic *\)

    1. \(l \homotopic *\)

30. \(M \homotopic *\)

    1. \(m \homotopic *\)

31. \(N \homotopic *\)

    1. \(n \homotopic *\)

32. \(O \homotopic S^1\)

    1. \(o \homotopic S^1\)

33. \(P \homotopic D \homotopic S^1\)

    1. \(p \homotopic P \homotopic S^1\)

34. \(Q \homotopic O \homotopic S^1\)

    1. \(q \homotopic p \homotopic o \homotopic S^1\)

35. \(R \homotopic D \homotopic S^1\).

    1. \(r \homotopic l \homotopic S^1\)

36. \(S \homotopic *\)

    1. \(s \homotopic S \homotopic *\)

37. \(T \homotopic *\)

    1. \(t \homotopic l \homotopic *\)

38. \(U \homotopic *\)

    1. \(u \homotopic U \homotopic *\)

39. \(V \homotopic *\)

    1. \(v \homotopic V \homotopic *\)

40. \(W \homotopic *\)

    1. \(w \homotopic W \homotopic *\)

41. \(X \homotopic *\)

    1. \(x \homotopic X \homotopic *\)

42. \(Y \homotopic *\)

    1. \(y \homotopic v \homotopic *\)

43. \(Z \homotopic *\)

    1. \(z \homotopic Z \homotopic *\)

This results in a partition of the alphabet into the following homotopy types:

• \(\theset{A, D, O, P, Q, R, S^1}\) \(\union \theset{a,b,d,e,g,o,p,q}\)

• \(\theset{C, E, F, G, H, I, J, K, L, M, N, S, T, U, V, W, X ,Y, Z, *}\) \(\union \theset{c,f,h,k,l,m,n,r,s,t,u,v,w,x,y,z}\)

• \(\theset{B, S^1 \vee S^1}\)

• \(\theset{i, j, \theset{*, *}}\)

Homeomorphisms: ignore ligatures!!

1. \(\theset{A, R}\) Can remove a point to obtain two components homeomorphic to \(\theset{I, F}\) respectively.

2. \(\theset{D, O, S^1}\) These all have no single point that can be removed to disconnect the space.

3. \(\theset{B,S^1 \vee S^1}\) Remove point at junction

4. \(\theset{C, G, I, J, L, M, N, S, U, V, W, Z, [0,1]}\) These all have a point that can be removed to yield two components, but no points that yield three. (Intuitively, all can be obtained by twisting a straight wire.)

5. \(\theset{E, F, T, Y, \bigvee_{i=1}^3 [0,1]}\) These all have a point that can be removed to yield 3 connected components homeomorphic to \(I\). This is the “pasting” point in the vee.

6. \(\theset{H, K, \bigvee_{i=1}^5 [0,1]}\) Can remove two points to disconnect each into five components.

7. \(\theset{P, Q, S^1 \vee [0,1]}\) Both contain a nontrivial loop.

8. \(\theset{X, \bigvee_{i=1}^4 [0,1]}\) Can remove one point to separate into four components.