THE HOMOTOPY TYPE OF THE SPACE OF RATIONAL FUNCTIONS

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1 THE HOMOTOPY TYPE OF THE SPACE OF RATIONAL FUNCTIONS by M.A. Guest, A. Kozlowski, M. Murayama and K. Yamaguchi In this note we determine some homotopy groups of the space of rational functions of degree d on the Riemann sphere, and describe a homogeneous space structure for the case of degree. As an application we show that C -structure on d 0 Hol d is not compatible with that on Ω S, where Hol d denotes the space of all based holomorphic maps of degree d from Riemann sphere to itself Mathematics subject classification: 55P15, 55Q5 1. Introduction. For each positive integer d, let Hol d denote the space of all holomorphic (equivalently, algebraic) maps of degree d from the Riemann sphere S = C to itself. This space is of interest both from a classical and a modern point of view (see [1], [5]). Let Hol d be the subspace of Hol d consisting of maps which preserve a basepoint of S. It is well known that Hol 1 is the group of fractional linear transformations PSL (C) and that Hol 1 may be identified with the affine transformation group of C. It is an elementary fact that Hol d and Hol d are connected spaces. The fundamental groups of these spaces are Z/d, Z respectively; these computations are due to Epshtein ([6]) and Jones (see [8]). The following more general result was obtained by Segal: Theorem 0 ([8]). Let Map d be the space of all continuous maps of degree d from S to itself and let Map d be the subspace consisting of maps f such that f( ) =1. Then the natural inclusion maps induce the following isomorphisms of homotopy groups: (1) If k<d, then π k (Hol d)=π k (Map d)=π k+ (S ). () If k<d, then π k (Hol d )=π k (Map d ). The stable homotopy type of Hol d was studied in [3]. In this note we shall extend the above results by determining some further homotopy groups of the space Hol d. Our results are as follows: 1

2 Theorem 1. (1) For k, π k (S 3 ) d=1 π k (Hol d )= π k (S 3 ) π k (S ) d= Z/ d 3,k = () If k 3 and d 3, then π k (Hol d )=π k (Hol d) π k (S 3 ). (3) In particular, if d>k 3, then π k (Hol d )=π k+ (S ) π k (S 3 ). Theorem. The space Hol may be identified with a homogeneous space of the form (SL (C) SL (C))/H, where H is isomorphic to C Z/4. In this semi-direct product, the action of Z/4 =<σ:σ 4 =1>is given by σ α = α 1 for α C. In particular, Hol is homotopy equivalent to (S 3 S 3 )/(S 1 Z/4). Theorem 3. (1) The universal cover of Hol is homotopy equivalent to S. () The universal cover of Hol may be identified with a homogeneous space of the form (SL (C) SL (C))/D, where D is isomorphic to C. In particular, it is homotopy equivalent to S 3 S. In Theorem 1, the case d = 1 follows from the fact that Hol 1 may be identified with PSL (C) and hence is homotopy equivalent to RP 3 ; the case d = is direct consequence of () of Theorem 3. In section, we shall consider the homogeneous structure of Hol based on the action of Hol 1 Hol 1 by pre- and post-composition, and give the proof of Theorem and () of Theorem 3. In section 3, we shall investigate the space Hol, and give the proof of (1) of Theorem 3. In section 4, we shall prove Theorem 1. In section 5 we shall give an application of these results to the C -operad structure on d 0 Hol d. In particular, we shall show that the C -structure on d 0 Hol d is not compatible with that on Ω S up to homotopy.. The Homogeneous Structure of Hol. From now on, we identify Hol d with the space of functions f = p 1 /p, where p 1,p are coprime polynomials such that max{deg(p 1 ), deg(p )} = d. The group Hol 1 acts on Hol d by pre- and post-compositions: for (A, B) Hol 1 Hol 1 and f Hol d we have (A, B) f (z) =A(f(B 1 (z)). The following proposition is well known, but we shall give a proof for the sake of completeness.

3 Proposition.1. The group Hol 1 Hol 1 acts transitively on Hol. Proof. Let f = p/q Hol. It suffices to show that A(f(B(z))) = z for some A, B Hol 1. Since Hol 1 acts transitively on S, there is a function A Hol 1 such that A(f( )) =. Hence, without loss of generality, we may suppose that f( ) =, i.e. that deg(p) => deg(q). Claim: If f( ) =, then there is some (A, B) Hol 1 Hol 1 such that A(f(B(z))) = z or (z + z 1 )/. We shall prove this by considering separately the cases deg(q) =0,deg(q) =1. (i) If deq(q) = 0, we may suppose that f(z) =p(z)=a(z+b) +cfor some a 0,b,c C. If we put A(z) =a 1 (z c), B(z) =z bthen A(f(B(z))) = z, as required. (ii) If deg(q) = 1, we may suppose that q(z) =z+a,andp(z)=b{(z+a) +c(z+a)+d } with b 0,d 0. Putting A(z) =(z bc)/bd, B(z) =dz a we see that A(f(B(z))) = (z + z 1 )/. This completes the proof of the claim. Let g(z) = (z + z 1 )/. If A(z) = (z +1)/(z 1), B(z) = (z 1)/(z + 1) then A(g(B(z))) = z. This completes the proof of Proposition.1. Remark: It follows from the claim that Hol consists of two Hol 1 Hol 1 orbits. It is well known that the map ( ) a b SL (C) Hol 1, az + b c d cz + d, is a double covering and induces an isomorphism PSL (C) = Hol 1 of Lie groups. Thus the group G = SL (C) SL (C) acts (transitively) on Hol. Lemma.. Let H denote the isotropy subgroup of SL (C) SL (C) at z Hol. Then H = {( ( ) ( )) ( ( ) ( )) } α 0 α 0 0 iα 0 α ± 0 α, 0 α 1, ± iα, 0 α 1 : α C. 0 Proof. This follows by direct calculation. Next we determine the group structure of H. Lemma.3. Let K = C Z/4 be the group defined by the action of Z/4 =<σ:σ 4 =1> on C by σ α = α 1 for α C. Then H and K are isomorphic Lie groups. Proof. We put σ 1 = ( ) 0 i, σ i 0 = ( )

4 Define ϕ : K H by (( ) ( ) (α, σ m α 0 ) 0 α σ1 m α 0, 0 α 1 σ m It is easy to check that ϕ is an isomorphism. Proof of Theorem. The first part of Theorem follows from Proposition.1, Lemma. and Lemma.3. The inclusion map of the maximal compact subgroup SU() = S 3 of SL (C) induces the homotopy equivalence (S 3 S 3 )/(S 1 Z/4) Hol. Next we consider the universal cover of Hol. Let D be the subgroup {(( ) ( )) } α 0 α 0 0 α, 0 α 1 : α C of SL (C) SL (C), and let D c be its maximal compact subgroup {(( ) ( )) } α 0 α 0 0 α, 0 α 1 : α =1,α C. Then D is a normal subgroup of H = C Z/4 isomorphic to C. Similarly D c is is isomorphic to S 1. Clearly the projection E = (SL (C) SL (C))/D (SL (C) SL (C))/H = Hol is a regular covering. To show that E is simply connected, we shall consider the fibre bundle SL (C) E SL (C)/D whose projection map is induced by the projection onto the second factor. Lemma.4. (1) E is fibre homotopy equivalent to the fibre bundle SU() Y S, where Y = (SU() SU())/D c. () Y can be identified with the unit sphere bundle S(η η ), where η denotes the Hopf complex line bundle over S = CP 1. ). Proof. (1) The inclusion map SU() SU() SL (C) SL (C) induces the desired fibre homotopy equivalence Y E. () The second factor S = {( ) } α 0 0 α 1 : α =1,α C of D c is the standard embedding of S 1 into SU(), and S 1 SU() SU()/S S is the Hopf bundle. We may use the identifications SU() = Sp(1) = S 3 H, and extend the action of D c naturally to SU() H. By considering the transition functions of the vector bundle (SU() H)/D c, it is not difficult to see that Y is equivalent to S(η η ). Proof of () of Theorem 3. It follows from the homotopy exact sequence that Y (and hence E) is simply connected. Hence E is the universal covering of Hol. Since π (BU()) = Z, a -dimensional complex vector bundle ξ over S is determined by its first Chern class c 1 (ξ). As c 1 (η η ) = 0, it follows that η η is trivial. Hence E Y S 3 S. 4

5 3. The Space Hol. In this section we shall use the actions of Hol 1 (and Hol 1) on Hol d by post-composition: A f(z) =A(f(z)) for (A, f) Hol 1 Hol d. First, we have two easy lemmas: Lemma 3.1. Let d 1. The group Hol 1 acts freely on Hol d by post-composition. Similarly, Hol 1 acts freely on Hol d by post-composition. Proof. This follows immediately from the fact that any map of non-zero degree is surjective. Lemma 3.. Let d 1. Then the natural inclusion map j d : Hol d Hol d induces a homeomorphism j d : Hol 1 \ Hol d Hol 1 \ Hol d. Proof. Since Hol 1 acts transitively on S, the induced map j d : Hol 1 \ Hol d Hol 1 \ Hol d is surjective. Since (Hol 1 f) Hol d = Hol 1 f for any f Hol d, j d is injective. If we identify these spaces by j d, it is easy to see that the topologies coincide. Proposition 3.3 ([4]). There is a fibration S 1 Hol RP. Remark: Cohen and Shimamoto ([4]) deduce this from results of Donaldson ([5]) and Atiyah and Hitchin ([1]) on monopoles. We shall give a direct and elementary proof. Proof. By Lemma 3.1 we have a principal bundle Hol 1 Hol Hol 1 \ Hol. By Theorem and Lemma 3., Hol 1 \ Hol Hol 1 \ Hol (S 3 /S 1 )/{±1} S /{±1} = RP. Since Hol 1 S 1, we have the required fibration S 1 Hol RP. Proof of (1) of Theorem 3. Consider the above fibration S 1 Hol π RP. Let p : S RP and q : X Hol be the universal coverings. Since X is simply connected, there is a lift θ : X S such that p θ = π q. It follows by diagram chasing that θ : π k (X) π k (S ) is an isomorphism for all k. Hence θ is a homotopy equivalence. 4. The Proof of Theorem 1. Let ι n π n (S n ) be the oriented generator and η π 3 (S ) be the class of the Hopf map. We put η n =Σ n η π n+1 (S n )forn>. The following three results are well known and we omit the proofs. 5

6 Lemma 4.1 ([9]). (1) π n (S n )=Z{ι n }. () π 3 (S )=Z{η },π n+1 (S n )=Z/{η n }for n>. (3) π n+ (S n )=Z/{ηn}for n>1.hereweputηn=η n η n+1. Lemma 4. ([7],[9]). (1) [ι,ι ]=η. () If k>,[ι,α]=0for any α π k (S ). Here [, ] denotes the Whitehead product. Let Map(S n,x) denote the space of all continuous maps from S n to X, and let Map (S n,x) be the subspace consisting of based maps. For a map f we denote by Map f (S n,x)ormap f(s n,x) the path-component containing f. Lemma 4.3 ([10]). Let f Map (S n,x) and let Map f (S n,x) Map f (S n,x) ev X be the evaluation fibration. If we use the identification π k (Map f (S n,x)) = π k+n (X), then the boundary operator : π k+n (X) π k 1 (X) of the homotopy exact sequence associated with the evaluation fibration is given (up to sign) by the Whitehead product: (α) =[α, f]. Proof of Theorem 1. (1) It suffices to consider the case d 3. Let I : Hol d Map d and J : Hol d Map d be the inclusion maps. By Theorem 0, I and J are homotopy equivalences up to dimension d. Consider the following commutative diagram: Hol ev d Hol d S I J = Map d Map d ev S in which the horizontal sequences are evaluation fibre sequences. The result follows from the induced diagram of homotopy groups, by using Lemmas 4.1, 4., and 4.3 and the Five Lemma. This completes the proof of (1). () Suppose that d 3andk 3 are integers. Consider the commutative diagram of principal bundles Hol 1 Hol p d d Hol 1 \ Hol d j d j d q d Hol 1 Hol d Hol1 \ Hol d where j d is a natural inclusion map. In the induced homotopy exact sequences, since Hol 1 S 1, (p d ) : π k (Hol d) π k (Hol 1 \ Hol d) is an isomorphism for k 3. Hence (j d ) (p d ) 1 ( j d ) 1 : π k (Hol 1 \ Hol d ) π k (Hol d ) gives a splitting of (q d ).Sowehave π k (Hol d )=π k (Hol d) π k (Hol 1 ). 6

7 Because Hol 1 RP 3, π k (Hol 1 )=π k (S 3 ) and this completes the proof of (). (3) It follows from Theorem 0 that π k (Hol d)=π k+ (S )fork<dand the assertion easily follows from (). Remark: The above method allows one to deduce the result of Epshtein ([6]) that π 1 (Hol d )= Z/dfrom the result of Jones (see [8]) that π 1 (Hol d)=z. 5. The C -operad structure on d 0 Hol d. Consider d 0 Hol d, the disjoint union of the based rational functions of degree d. It is known that this is a C -operad space ([]). Let µ d : F (C,d) (S 1 ) d Hol d be the structure map, where we identify Hol 1 up to homotopy with S 1. Let i d : Hol d Ω d S and i : Hol Ω S be the inclusion maps. It is known that i is a C -map up to homotopy ([]). It follows from the May-Milgram model of Ω Σ X that we can identify Ω S 3 with J(S 1 ), where J(X) denotes the space J(X) = d 1 (F(C,d) Σd X d )/ and is a well known equivalence relation. On the other hand, there is a well known equivalence Ω S 3 Ω 0S. The following observation of Professor F.R. Cohen shows that the C -structure on Ω 0S is incompatible with the one on d 0 Hol d. (This contradicts the statement of [3] that diagram (3.4) of that paper is homotopy commutative.) Let J d (X) bethed-th term of the May-Milgram filtration on J(X). 7

8 Proposition 5.1. There is no homotopy equivalence θ :Map (S,S )=Ω S Ω S 3 such that the following diagram is homotopy commutative: F (C, ) Σ (S 1 ) µ Hol i Ω S j θ J (S 1 ) J (S 1 ) Ω S 3 Here j denotes the inclusion map. Proof. Suppose that the above diagram is homotopy commutative. Since F (C, ) S 1, there is a non-zero element e 1 e 1 H 3 (F (C, ) Σ (S 1 ), Z/) = Z/ which represents a generator. Using the above diagram, if we put α =(µ ) (e 1 e 1), we have 0 Q 1 (e 1 )=(θ i ) (α) H 3 (Ω S 3, Z/). Since Q 1 (e 1 ) H 3 (Ω S 3, Z/) is primitive, and θ is an equivalence, the element (i ) α = θ 1 (Q 1 (e 1 )) H 3 (Hol, Z/) = Z/ is also primitive (the image of a primitive element is primitive). By [8], (i ) is injective, so α is primitive. Thus the generator of H 3 (Hol, Z/) = Z/ is indecomposable. On the other hand, because the universal cover of Hol is homotopy equivalent to S and π 1 (Hol )=Z, there is a fibration S Hol BZ S 1. Consider the mod Serre spectral sequence of this fibration. This collapses at the E level, and the generator of H 3 (Hol, Z/) = Z/ is decomposable. This is a contradiction. Remark: The above result implies that the C -structure of Hol d andthatofω S 3 are not compatible (up to homotopy) at least when d =. This was also pointed out in [4]. Acknowlegements: The first, second and fourth authors are indebted to the Mathematics Department of Tokyo Institute of Technology for its hospitality. The application to C - operad structures is due to Professor F.R. Cohen. A gap in an earlier version of the proof of Theorem 3 was filled thanks to Professor A. Kono, who suggested the present version. The authors are grateful to Professors Cohen and Kono for their kind assistance. 8

9 References [1] M.F. Atiyah and N.J. Hitchin, The Geometry and Dynamics of Magnetic Monopoles, Princeton Univ. Press, [] C.P. Boyer and B.M. Mann, Monopoles, non-linear σ models and two-fold loop spaces, Commun. Math. Phys. 115 (1988), [3] F.R. Cohen, R.L. Cohen, B.M. Mann and R.J. Milgram, The topology of rational functions and divisors of surfaces, Acta Math. 166 (1991), [4] R.L. Cohen and D.H. Shimamoto, Rational functions, labelled configurations and Hilbert schemes, J. Lond. Math. Soc. 43 (1991), [5] S.K. Donaldson, Nahm s equations and the classification of monopoles, Commun. Math. Phys. 96 (1984), [6] S.I. Epshtein, Fundamental groups of spaces of coprime polynomials, Functional Analysis and its Applications 7 (1973), [7] P.J. Hilton and J.H.C. Whitehead, Note on Whitehead product, Annals of Math. 58 (1953), [8] G.B. Segal, The topology of spaces of rational functions, Acta Math. 143 (1979), [9] H. Toda, Composition Methods in Homotopy Groups of Spheres, Vol. 49, Princeton Univ. Press, 196. [10]G.W. Whitehead, On products in homotopy groups, Annals of Math. 47 (1946), Department of Mathematics, University of Rochester, Rochester, New York 1467, USA Department of Mathematics, Toyama International University, Kamishinkawa, Toyama, Japan Department of Mathematics, Tokyo Institute of Technology, Oh-Okayama, Meguro, Tokyo 15, Japan Department of Mathematics, The University of Electro-Communications, Chofu-gaoka, Chofu, Tokyo 18, Japan 9

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