By Lisa Jacobsen

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**Additional resources for Analytic Theory of Continued Fractions III: Proceedings of a Seminar Workshop**

**Example text**

These are both C" for 0 I t 5 1 sincef(x) # 0. Thus H(x, t ) = (1 - 2t)x is a homotopy in S" from 1 to a , . + 2Jt-tzf(x>/llf(x)II I Remark The converse is also true, but the technical details in proving it are harder and we will not need it to solve Problem 3. Essentially, given a homotopy from 1 to a , , one approximates this with a differentiable homotopy. Then the tangent line to the curve P,(t) = H ( x , t ) at t = 0 contains a unit vector pointing in the direction of increasing t, which is tangent to the sphere, and nonzero.

We give three examples: (1) J(J,+J(Jz> (2) JlGz --+%, (3) G + S . The functor is the identity on objects and maps, but considers them as different things. Thus, every R-module may be considered as an abelian group by forgetting the R-module structure. Every R-module homomorphism may be considered as a group homomorphism. Similarly for (2) and (3). Example 5 The identity functor from any category to itself. It is the identity on objects and maps and is covariant. 3 A natural transformation cp from T, to T,, where TI and T , are functors from a category C, to a category C, , written ~p:Ti-tTz, 4 .

3 Ym 7 . . , r k , s,, . . , s l , andfi(zi)f2 (zi)-' for 1 I i 5 j . 7. Calculating the Fundamental Group 44 Proof Let G be the group defined by these generators and relations above. One can clearly find a map cp: G 4 GI *c G, such that cp(xi) = x i , and cp(yi)= y i since these relations hold in C , * G G, . On the other hand, there are maps hi : G i -+ G with hlfi = h, f 2 given by h,(xi) = xi, h,(y,) = y i . 13(a) there is a map h : G, *c G, G with h(xi) = x i and h(yi) = y i . Clearly hcp = 1 and cph = 1 since these composites are the identity on a set of generators.