View of Topological equivalence of finitely determined real analytic plane-to-plane map germs

TOPOLOGICAL EQUIVALENCE OF FINITELY DETERMINED REAL ANALYTIC

PLANE-TO-PLANE MAP GERMS

OLAV SKUTLABERG^∗

Abstract

Generic smooth map germs(R²,0)→(R²,0)are topologically equivalent to cones of mappings S¹ →S¹. We carry out a complete topological classification of smooth stable mappings of the circle and show how this classification leads, via the result mentioned above, to a topological classification of finitely determined real analytic map germs(R²,0)→(R²,0).

1. Introduction

Let f and g be smooth mappings between smooth manifolds N and P of dimensionsnandp, respectively. Let 0 ≤ r ≤ ∞. We say thatf andg are Ar-equivalent if there is a commutative diagram

N −−−−−→^f P

h↓

↓^k N −−−−→^g P

wherehandkareC^r diffeomorphisms. Similarly, iff andgare smooth map germs(N, p)→(P , q), then we say thatf andgareA^r-equivalent if there is a commutative diagram

(N, p)−−−−−→^f (P , q)

↓

h ↓^k

(N, p)−−−−→^g (P , q)

wherehandk are germs ofC^r diffeomorphisms.A0-equivalence is usually referred to as topological equivalence. Let C^∞(N, P )be the set of proper

∗The author wishes to thank Hans Brodersen for ideas and corrections, and Magnus D. Vigeland for suggesting the proof of Proposition 2.9.

Received 25 February 2010, in final form 7 October 2010.

smooth mappingsN → P, and letC^∞(n, p) (resp.O(n, p)) be the set of smooth (resp. real analytic) map germs(Rⁿ,0)→(R^p,0).

A subset⊂C^∞(n, p)(resp.O(n, p)) isproalgebraicif

=

r≥1

(j^r)⁻¹(r)

where eachr ⊂J^r(n, p)is an algebraic subvariety. A proalgebraic setis ofinfinite codimensionif

rlim→∞cod_r = ∞.

A property of smooth (real analytic resp.) germs is said to holdin generalif the set of germs not having the property is contained in a proalgebraic set of infinite codimension.

By the cone of a smooth mapf : Sⁿ⁻¹ → S^p−1, we mean the mapF : Sⁿ⁻¹×[0,1)/Sⁿ⁻¹× {0} →S^p−1×[0,1)/S^p−1× {0}given by

F ([(p, t )])=[(f (p), t )].

Consider the spaceC^∞(n, p)whenn≤p,n =4,5 and(n, p)is in the ‘nice range’. The ‘nice range’consists of the pairs of dimensions ofNandPsuch that the set of proper smooth stable mappingsN →P is dense in the set of proper smooth mappingsN →P. It is shown in [2] that for germs inC^∞(n, p), the property of having a realization which is topologically equivalent to the cone of a smooth stable mappingSⁿ⁻¹ → S^p−1 via homeomorphisms which are diffeomorphisms outside the origin holds in general. We say that map-germs with this property aregeneric. Thus, forn, pin this range, the classification of generic map germs(Rⁿ,0)→(R^p,0)with respect to topological equivalence is contained in the classification of the smooth stable mappingsSⁿ⁻¹→S^p−1 in the sense that theA0-equivalence class inC^∞(n, p)of a generic map germ corresponds to anA∞-equivalence class inC^∞(Sⁿ⁻¹, S^p−1).

In this paper we carry out this classification in the real analytic case for n=p=2. In Section 2 we classify the smooth stable mappingsS¹→S¹and show how to generate complete lists of theA∞-equivalence classes of such mappings. In the case of 1-dimensional spheres, the classification is essentially a combinatorical problem. In Section 3 we classify finitely determined real analytic map germs(R²,0)→(R²,0)using the above strategy. Our method solves the so-called ‘recognition problem’: Given two finitely determined real analytic map germs(R²,0)→(R²,0), are theyA0-equivalent?

Some of the results in this article have been obtained independently by Moya-Pérez and Nuño-Ballesteros in [6].

2. Classification of smooth stable mappingsS¹→S¹

In this section we define invariants giving a complete classification of smooth stable mappingsS¹ → S¹. Letf : S¹ → S¹be a smooth stable mapping.

Thenf has only Morse singularities, (f )is finite and f has no singular double points.

2.1. Definition ofAst(f )

LetP : [0,2π )→S¹be the parametrization given byP (t )=(cost,sint ). If f has no singular points, then we define Ast(f )= (p, p, . . . , p)wherepis repeated #f⁻¹(1)times. Assumef has singular pointss_i(f ),i=1, . . . , n(f ) where k < l ⇒ P⁻¹(s_k(f )) < P⁻¹(s_l(f )). Let σ_i(f ) = f (s_i(f )) and let f⁻¹(σ_i(f ))\ {s_i(f )} = {p_ij(f )}j^m=ⁱ1 where k < l ⇒ P⁻¹(p_ik(f )) <

P⁻¹(p_il(f )). Let

A(f )= {a1, a2, . . . , a_N} =P⁻¹ n

i=1

{s_i(f )} ∪ {p_ij(f )}j^m=ⁱ1

wherei < j ⇒ai < aj andN = N (f )= n(f )+n(f )

i=1 mi. Let(f ) = f ((f ))= {σ1(f ), . . . , σ_n(f )}and define

B(f )= {b₁, b₂, . . . , b_{n(f )}} =P⁻¹((f )) wherei < j ⇒bi < bj.

Next, let

S= {s, p}, S^∗= ∞ i=1

{s_i, p_i}

and define mapsT :A(f )→SandT^∗:A(f )→S^∗given by T (x)= s, ifP (x)=s_i(f );

p, ifP (x)=p_ij(f ), T^∗(x)= s_i, ifP (x)=s_i(f );

p_i, ifP (x)=p_ij(f ). Now, define theassociated tuplesoff to be the orderedN (f )-tuples

Ast(f )=

T (a1), T (a2), . . . , T (aN (f ))

and Ast^∗(f )=

T^∗(a1), T^∗(a2), . . . , T^∗(aN (f ))

Remark2.1. Given Ast^∗(f )one can obtain Ast(f )by just forgetting the indices of thesandpin Ast^∗(f ). Conversely, given Ast(f ), it is easy to find the right indices for thesin Ast^∗(f )and then we can find the indices ofpin

p₁₁

p₂₁

p₃₁ p₄₁

s₁

f

c(t) s₂

σ₂ σ₁

σ₃ σ₄

s₃ s₄

Figure1. Visualization of a mapf :S¹→S¹with Ast(f )=(p, s, s, p, p, s, s, p). The curvec: [0,2π )→^R2is such thatc(t )/c(t ) =f (P (t )).

Ast^∗(f )as well, using the fact that at a singular point,f changes the behaviour of being orientation preserving or orientation reversing. This enables us to find the correct indices ofp.

2.2. Legal permutations

LetS_k be the group of permutations ofZ/ kZ. Some permutations are of par- ticular interest when trying to classify stable maps underA0-equivalence. We start with some definitions.

Definition2.2. An elementσ ∈S_kis aswitchif there is somea∈^Zsuch that

σ ([x])=[x+a].

Let Swkbe the set of switches inS_k.

Definition2.3. The permutationr∈S_kgiven byr([x])=[−x] is called thereversation. LetRk = {id, r}.

Definition2.4. The subgroupLk = {σ ◦τ |σ ∈ Swk, τ ∈Rk}ofSk is called the group oflegal permutations.

Let X be a set. For everyk, lete_k : {1,2, . . . , k} → ^Z/ kZbe the bijec- tionx → [x]. We introduce an equivalence relation E_k onX^k by the rule (t1, t2, . . . , tk) ∼ (t₁, t₂, . . . , t_k) if there is a permutation ρ ∈ Lk such that ti =t

e⁻¹_k (ρ(ek(i)))for alli=1, . . . , k. Denote theEk-equivalence class oft ∈X^k by [t]E. Forρ∈S_kandt ∈X^k, letρ·t ∈X^kbe defined by(ρ·t )_i =t_e−1

k (ρ(ek(i)))

fori =1, . . . , k. For simplicity we writeρ(i)fore⁻_k¹(ρ(ek(i))).

2.3. The main theorem of the classification

The aim of this section is to prove the following theorem.

Theorem2.5.Letf, g ∈C^∞(S¹, S¹)beC^∞-stable. Then

f ∼A∞ g⇔N (f )=N (g) and [Ast(f )]E =[Ast(g)]E. Proof. We prove the theorem when(f ), (g) = ∅. The same technique applies when(f )=(g)= ∅. The theorem is proved in three steps:

Step 1is to prove that Ast(f )=Ast(g)⇒f ∼A∞ g. Suppose Ast(f ) = Ast(g). After composition with diffeomorphisms in source, we may assume thatA(f )=A(g)and that(1,0)is a regular point off, and hence also ofg.

A priori, it may happen thatf is orientation preserving onP ([a1(f ), a2(f )]) while g is not, but after composition with a diffeomorphism in target, we may assume thatf and g are orientation preserving on the same subset of source, and thatσ_i(f )= σ_i(g)for allias well. Finally, we may assume that (1,0) /∈f ((f )).

We are going to define a smooth homotopyf_t of stable mappings ofS¹ starting atf and ending atg. Thef_t will be smoothly equivalent, and hence, f ∼A∞ g. The standard technique for producing homotopies between map- pings in Euclidean space by taking convex combinations of the mappings is not applicable here, sinceS¹is not a vector space. Nevertheless, by choosing appropriate charts, the same strategy may be applied to coordinate neighbour- hoods, and our assumptions onf andg ensure that the resulting mapping is in fact a smooth homotopy. The details are as follows.

Letn = n(f ) = n(g), and letN = N (f ) = N (g). Letτ ∈ Sn be such that bi = bi(f ) = P⁻¹(στ (i)(f )). Notice thatb1 > 0 by the assumption (1,0) /∈(f ). Let

0< v < 1 2min

i (bi+1−bi, b1,2π−bn) and define

_i :P (b_i−v, b_i+1+v)→(−v, b_i+1−b_i+v), P (x)→x−b_i fori =1, . . . , n−1. Fori=nwe define

n:S¹\P ([b1+v, bn−v])→(−v, b1+2π−bn+v) by

P (x)→ 2π−b_n+x, x∈[0, b1+v) x−b_n, x∈(b_n−v,2π ).

The mappings_i, i=1, . . . , nare well defined by the choice ofv. Together with their domains of definition, they coverS¹with local charts.

Let

0< u < 1 2min

i (a_i+1−a_i, a₁,2π−a_N)

and letUi =P (ai−u, ai+1+u), i =1, . . . , N −1 andUN =S¹\P ([a1+ u, a_N−u]). In the same way, letV_i =P (b_i −v, b_i+1+v), i =1, . . . , n−1 andV_n = S¹−P ([b1+v, b_n−v]). We can now define our homotopy. By continuity off and g, if uis small enough, then for alli there is a j such that bothf (U_i)andg(U_i)are contained inV_j. More precisely; there exists ρ : {1, . . . , N} → {1, . . . , n} such that for all i, f (Ui)∪g(Ui) ⊂ Vρ(i). For even smalleru, we can ensure that cl(f (Ui)∪g(U_i)) ⊂ V_ρ(i). Let F : S¹×(− ,1+ )→S¹be defined by

F (p, t )=f_t(p)

=⁻_ρ(i)¹(t _ρ(i)(g(p))+(1−t )_ρ(i)(f (p))), p∈U_i.

We need to show thatf_t(p)is well defined onS¹and thatf_t(p)is smooth. The continuity ofρ(i)and the observation thatt ρ(i)(g(p))+(1−t )ρ(i)(f (p)) lies between_ρ(i)(g(p))and_ρ(i)(f (p))for 0≤t ≤1, shows thatf_tis well defined onU_i when is chosen small enough.

Next we show that the definitions off_t agree onU_i ∩U_j. It is enough to check the combinations(i, j ) = (N,1)and (i, j ) = (i, i+1)fori < N. The other combinations ofi and j giveUi ∩Uj = ∅. We first assume that 1≤ρ(i)=ρ(j )−1< n. Writing out the definitions,

⁻_ρ(i)¹(t ρ(i)(g(p))+(1−t )ρ(i)(f (p)))

=⁻_ρ(i)¹ (t[P⁻¹(g(p))−bρ(i)]+(1−t )[P⁻¹(f (p))−bρ(i)])

=⁻_ρ(i)¹ (t[P⁻¹(g(p))]+(1−t )[P⁻¹(f (p))]−b_ρ(i))

=P (t[P⁻¹(g(p))]+(1−t )[P⁻¹(f (p))])

=⁻_{ρ(j )}¹ (t _{ρ(j )}(g(p))+(1−t )_{ρ(j )}(f (p))).

Forρ(i)=nandρ(j )=1, we have

⁻₁¹(t 1(g(p))+(1−t )1(f (p)))

=P (t[P⁻¹(g(p))]+(1−t )[P⁻¹(f (p))])

and

⁻_n¹(t n(g(p))+(1−t )n(f (p)))

=⁻_n¹(t[P⁻¹(g(p))+2π−b_n]+(1−t )[P⁻¹(f (p))+2π−b_n])

=⁻_n¹(t[P⁻¹(g(p))]+(1−t )[P⁻¹(f (p))]+2π−b_n)

=P (t[P⁻¹(g(p))]+(1−t )[P⁻¹(f (p))]).

This shows thatf_t is well defined onS¹in this case. The case 1 < ρ(i) = ρ(j )+1≤nand the caseρ(i)= 1, ρ(j )= nmay be checked in a similar way.

It remains to show that f_t has finitely many singularities, all of Morse type, and no singular double points. In fact,f_t has the same singular set and discriminant set asf, g. To actually show this, we need to work with charts in the source too. Let

θ_i :U_i →(−u, a_i+1−a_i+u), P (x)→x−a_i fori =1, . . . , N−1. Fori=N we define

θN :UN →(−u, a1+2π−aN +u) by

P (x)→ 2π−aN +x, x ∈[0, a1+u) x−aN, x ∈(aN −u,2π ).

The charts(θ_i, U_i)coverS¹. Now, we may compute

(2.1) ρ(i)◦f_t◦θ_i⁻¹(x)=t _ρ(i)(g(θ_i⁻¹(x)))+(1−t )_ρ(i)(f (θ_i⁻¹(x))).

By our assumptions,f andgare equally oriented at every regular point, and therefore the derivatives with respect tox of two terms on the right side of (2.1) have the same sign, and hence,(f_t)= (f ) = (g). Moreover, by definition of Morse singularities, we must have

d²

dx²_ρ(i)(f (θ⁻¹(x))) =0 and d²

dx²_ρ(i)(g(θ⁻¹(x))) =0 wheneverθ⁻¹(x) ∈ (f )and these second derivatives must have the same sign at singular points. It follows that in these charts, the second derivative of f_twith respect toxis different from 0 at every singular point. Therefore,f_thas only Morse singularities. From the definition off_t, we see thatf (p)=g(p) implies thatf (p)=g(p) =f_t(p). It follows thatf_t has no singular double points, and hence,ft is stable.

Step 2. Assume that Ast(f ) = ρ·Ast(g)for someρ = σ ·τ ∈ L_{N (g)}, whereσ ∈SwN (g),τ ∈R_{N (g)}. Ifτ = id, then there is someθ = θ (σ )such that if

Rσ :S¹→S¹ is given by

e^iθ →e^{i(θ+θ (σ ))}, then

Ast(f )=Ast(g◦Rσ)

and by Step 1 there are diffeomorphisms h and k such that the following diagram commutes.

S¹ S¹

S¹

S¹ S¹

g Rσ

g◦Rσ

h

f

k

Similarly, ifσ =id andτ =r ∈R_{N (g)}, then, if M :S¹→S¹ is given by

e^iθ →e^−iθ, then

Ast(f )=Ast(g◦M) and by Step 1 again, we have a commutative diagram:

S¹ S¹

S¹

S¹ S¹

g RM

g◦M

h

f

k

Ifρ =σ ◦r for someσ ∈SwN (g), then

Ast(f )=σ·Ast(g◦M)=Ast(g◦M◦R_σ),

which again, by the above arguments, implies thatf ∼A∞ g. Altogether we have shown that [Ast(f )]E =[Ast(g)]E ⇒f ∼A∞ g.

Step 3. Suppose thatf and g areA∞-equivalent. Then there are diffeo- morphismshandkofS¹such thatk◦f =g◦h. Since a singularity of Morse type is topologically different from a regular germ, it is clear thath maps (f )to(g), and thatkmaps(f )to(g), and it follows thatf⁻¹((f )) is mapped ontog⁻¹((g))byh, and hence, N (f ) = N (g). If his orient- ation preserving and h(s1(f )) = si(g), then Ast(g) = ρ · Ast(f ) where ρ([j])=[j+i−1]. Ifhis orientation reversing, then Ast(g)=ρ·Ast(f ) whereρ([j])=[i−j+1]. It follows that [Ast(f )]E =[Ast(g)]E.

2.4. Feasible tuples

By Theorem 2.5, the problem of listing all topological equivalence classes of smooth stable mapsS¹ → S¹corresponds to the problem of listing allE_n- equivalence classes of associated tuples to such maps. Every non-singular map f :S¹→S¹is clearly equivalent to the mape^iθ →e^inθwheren=#f⁻¹(1).

To generate such a list for maps with singularities, we will make use of another version of our tuples. Iff :S¹→S¹is a smooth stable map with(f ) = ∅, then [Ast(f )]Emay be represented by a tuple in{s, p}^{N (f )}having ansas the last component. This may be done in several different ways. LetA be such a representation. Letρ : {1, . . . , n(f )} → {1, . . . , N (f )}be such thatρis increasing andA_ρ(i) = s. Setρ(0)=0 and letci = ρ(i)−ρ(i−1)−1 for i=1, . . . , n(f ). Define

Ast^#(A)=(c1, c2, . . . , c_{n(f )})∈^{Nn(f )}0 . Let Ast^#(f )=[Ast^#(A)]E

whereA_{n(f )} = s. It is not difficult to see that this definition of Ast^#(f )is unambiguous.

Remark2.6. Clearly, Theorem 2.5 is still valid for maps with singularities if we replace Ast with Ast^#.

Given an element(x₁, x₂, . . . , x_n)∈^Nn0, we want to determine whether or not there is a smooth stable mapf :S¹→S¹such that

Ast^#(f )=[(x1, . . . , xn)]E.

Letf be a stable map with Ast^#(f ) = [(x1, . . . , x_n)]E. We say thatf is of type(n, m)ifn(f )=nandN (f )−n(f )=m. Thus, iff is of type(n, m), thennis an even number and

x1+x2+ · · · +xn=m.

These two properties arise from observing thatf has an even number of sin- gular points, and thatN (f )−n(f )is the number of regular preimage points of the discriminant set. Another property offis thatfrestricted to its singular set is injective, and this fact should be reflected in [(x1, . . . , x_n)]E. Indeed, the curveP (x),x ∈[0,2π ), passesx_i points inf⁻¹((f ))\(f )whenxruns throughI = [P⁻¹(s_i−1), P⁻¹(s_i)). Therefore, the curvef (P (x))passesx_i singular values in the same interval of parameters. Thus, ifσi−1=P (bj)and f is orientation preserving onP (intI )and

k =(remainder of the divisionx_i byn)+1, thenσ_i =P (b_j+k).

In general, letR :Z→ {1,2, . . . , n}be given byR(x)=(remainder of the divisionxbyn)+1. Letτ ∈S_nbe as in the proof of Theorem 2.5, i.e. such that P (bi) = στ (i). Assuming that(x1, x2, . . . , xn) = Ast^#(Ast(f )),σ1 = P (bj) and thatf is orientation reversing onP (P⁻¹(s1), P⁻¹(s2)), then we see that

σk =σ_{τ (R(j−x}

1−1+k

i=1(−1)ⁱ⁺¹[xi+1]))

fork =1, . . . , n. Moreover, since we chose representatives withsin the last component in the definiton of Ast^#, we have

σ_n=P (b_{R(j−x1−1)}).

In order for all these equations to be satisfied, the set R =

k i=1

(−1)ⁱ⁺¹[xi+1];k =1, . . . , n

has to be a complete remainder system modulo n, i.e., the canonical map R→^Z/nZis surjective. Furthermore,

n i=1

(−1)ⁱ⁺¹[xi+1]≡0 mod n.

Definition 2.7. An elementA = (x₁, . . . , x_n) ∈ ^Nn0 is feasible of type (n, m)ifnis an even number and the following condtions are satisfied:

(1) n

i=1x_i =m.

(2) n

i=1(−1)ⁱ⁺¹xi ≡0 mod n.

(3) k

i=1(−1)ⁱ⁺¹[xi+1];k=1, . . . , n

is a complete remainder system modulon.

Remark2.8. There are no feasible tuples of type(n, m)ifmis odd, because the numbersm=n

i=1xi andn

i=1(−1)ⁱ⁺¹xi have the same parity, and by 2 in the definition, the latter number is even, sincenis even.

Proposition2.9. There are no feasible tuples of type (n, m) if n ≡ 0 mod 4andm≡2 mod 4.

Proof. Assume that (x₁, . . . , x_n) is feasible of type (n, m). Let L_k = k

i=1(−1)ⁱ⁺¹(xi +1). Notice that 2

n−1 i=1

(−1)ⁱ⁺¹L_i

−L_n=x₁+ · · · +x_n+n=m+n.

SinceL_n ≡0 modn,

(2.2) 2

n−1

i=1

(−1)ⁱ⁺¹L_i ≡m mod n.

Since{L_k;k =1, . . . , n}is a complete remainder system modulon, we have

(2.3) 2

n i=1

Li ≡2

n−1

i=0

i≡n(n−1)≡0 mod n.

Addition of (2.2) and (2.3) yields

(2.4) 4(L1+L3+L5+ · · · +Ln−1)≡m mod n.

Hence, there is an integerKsuch that

4(L1+L3+L5+ · · · +Ln−1)−m=Kn.

It follows that 4|n⇒4|m.

The next theorem justifies the term ‘feasible tuple’.

Theorem2.10. LetA∈^Nn0. There exists a smooth stable mapf :S¹→S¹ withAst^#(f )=Aif and only ifAis feasible of type(n, m)for some number m.

Proof. The forward implication follows from the above discussion. For the other implication, letA=(x₁, . . . , x_n)∈^Nn0be a feasible tuple of type(n, m).

We need to construct a smooth stable mapf : S¹ →S¹with Ast^#(f )= A.

We construct a smooth mapfA : [0,2π ) →^Rsuch thatf = P ◦fA◦P⁻¹ is smooth and stable and satisfies Ast^#(f ) = A. It is natural to definef_A to consist of line segments outside some small open intervals about the singular points and consist of a modified parabola around the singular points. This strategy calls for some kind of gluing process, but we can not use a standard partition of unity, because we must have full control over the singularities off˜, and a partition of unity might introduce unwanted singularites. Instead, we will constructf_A explicitly, using smooth “bump functions” to glue the different parts of the function together.

Let

j (x)= e^{−(x−1)−2}·e^−(x+1)−2, x∈(−1,1)

0, otherwise

and let

k(x)= x

−1j (t )dt 1

−1j (t )dt .

Define

l(x)=

⎧⎨

⎩

x, x≤ −1

x−2xk(x), x∈(−1,1)

−x, x≥1.

Then

l(x)=

⎧⎨

⎩

1, x ≤ −1

1−2k(x)−2xk(x), x ∈(−1,1)

−1, x ≥1,

and

l(x)=

⎧⎨

⎩

0, x≤ −1

−4k(x)−2xk(x), x∈(−1,1)

0, x≥1,

Sincek is flat at−1 and 1,l is aC^∞ function onR. Also,l is increasing for x ≤ 0 and decreasing forx ≥0. Sincel(0)= 0 andl(0)= −4k(0) <0, this means that l has its only extreme point at x = 0 and this is a global maximum and a Morse singularity. The definition off_Ais the following. For k=1, . . . , n, let

X_k = k

i=1

(x_i+1), Y_k = k

i=1

(−1)ⁱ⁺¹(x_i+1), J_k=

X_k− ¹₂, X_k+ ¹₂ .

Let

I0=₁

2, X1− ¹₂ and fork=1, . . . , n−1 let

Ik =

Xk+ ¹₂, Xk+1− ¹₂ .

For a setB ∈^R, letχBbe the corresponding characteristic function which is 1 onBand 0 elsewhere. PutX0=Y0=0. Fork =1, . . . , n, let

Fk(x)=

Yk−1+(−1)^k−1(x−Xk−1) χI_k−1

G_k(x)=

Y_k+ (−1)^k+1

2 l(2(x−X_k))

χ_Jk.

Let

H (x)= n

i=1

(F_k(x)+G_k(x)).

Finally, let

f_A(x)= 2π n H

Xn

2πx+ 1 2

.

With this definition offA, letf = P ◦fA ◦P⁻¹. It is messy, but straight- forward to see thatf is smooth and that Ast^#(f )=A.

Letf be a smooth stable map of the circle. All the topological properties off are coded in Ast^#. We show how|degf|can be retrieved from Ast^#(f ).

Proposition2.11. LetAst^#(f )=(x₁, x₂, . . . , x_n). Then

|degf| = 1

n n

i=1

(−1)ⁱ⁺¹x_i .

Proof. LetA=Ast^#(f ), and letf_A be as in the proof of Theorem 2.10.

Then deg(f )=deg(P◦f_A◦P⁻¹). Certainly,f_Ais homotopic tof˜_Agiven by f˜_A(x)= 1

n n

i=1

(−1)ⁱ⁺¹x_i

x.

by the homotopyF (x, t )=tf_A(x)+(1−t )f˜_A(x). Clearly,p◦ ˜f_A◦P⁻¹has degree ¹_nn

i=1(−1)ⁱ⁺¹x_i, and this finishes the proof.

2.5. Tables of feasible tuples

A complete classification of smooth stable mapsS¹ → S¹ can be given by listing all the feasible tuples up to legal permutations. This task is well suited for recursive computer programming. Table 1 and Table 2 give MATLAB generated lists of feasible tuples and numbers of topological types for different (n, m).

(n, m) Number of topo-

Feasible tuples logical types

(4,4) 2 (1,2,1,0),(2,0,2,0)

(4,8) 5 (5,2,1,0),(1,6,1,0),(2,4,2,0), (6,0,2,0),(4,1,2,1)

(9,2,1,0),(5,6,1,0),(1,10,1,0), (4,12) 12 (6,4,2,0),(2,8,2,0),(5,2,5,0),

(8,1,2,1),(4,5,2,1),(6,1,4,1), (10,0,2,0),(6,0,6,0),(4,2,4,2)

(6,6) 1 (2,0,2,0,2,0)

(6,8) 2 (3,1,0,3,1,0),(2,0,1,4,1,0) (6,10) 3 (3,0,4,2,1,0),(1,4,0,4,1,0),

(3,1,2,1,3,0) Table1. Table of topological types

(n, m) Number of topo-

(n, m) Number of topo-

logical types logical types

(4,16) 21 (8,8) 1

(4,20) 36 (8,12) 12

(4,24) 54 (8,16) 34

(4,28) 80 (10,10) 1

(6,12) 9 (10,12) 0

(6,14) 10 (10,14) 3

(6,16) 16 (10,16) 6

Table2. Number of topological types

Our tables lack the number of feasible tuples of type(2, m)because of the next proposition.

Proposition2.12. The number ofE₂-equivalence classes of feasible tuples of type(2, m)is_m

4

+1.

Proof. Assume(x₁, x₂)is feasible of type(2, m). Then x1+1≡1 mod 2

x1−x2≡0 mod 2.

These equations are satisfied if and only ifx₁is even andx₁andx₂have the same parity. The feasible tuples of type(2, m)are therefore

(2i, m−2i);i = 0,1, . . . ,^m₂

. There are ^m₂ + 1 elements in this set, and (2i, m− 2i) ∼E2

(m−2i,2i)for alli. Ifm=4kfor somek ∈^N, then ^m₂ +1=2k+1 is odd, and the number ofE2-equivalence classes isk+1= ^m₄ +1. Ifm=4k+2, then ^m₂ +1 = 2k+2 is even, and the number of equivalence classes is still k+1= ^m₄ +1.

3. Classification of finitely determined real analytic map germs(^R2,0)→(^R2,0)

LetO =O(2,2)be the set of real analytic map germs(R²,0)→(R²,0). Let Og = Og(2,2) ⊂ O(2,2) be the set of finitely determined map germs. By Theorem 0.5 of [7], finite determinacy holds in general inO(2,2).

3.1. Geometric properties

Finitely determined real analytic plane-to-plane germs have the following well known geometric properties.

Proposition3.1. For everyf ∈Ogthere is an open neighbourhoodU of 0 inR²and a real analytic representative off,fˆ:U →^R2such that

(1) fˆ⁻¹(0)= {0},

(2) fˆ|((f )ˆ \ {0})is injective,

(3) everyp∈(f )ˆ \ {0}is a fold point.

Proof. The proof of (2) and (3) goes as the proof of Lemma 6.2 in [1] with semialgebraic replaced by semianalytic. To prove (1), note thatfˆ⁻¹(0)\{0}is a semianalytic set. If 0 is in its closure, then by the Curve Selection Lemma, there is a real analytic curveγ : [0, )→^R2withγ (0)=0,γ (0, )∈ ˆf⁻¹(0)\{0}. Hence,fˆis identically 0 alongγ, but this contradicts both (2) and (3).

For the rest of this section, letf ∈Og, letU be a small ball around 0 and letfˆ : U → ^R2 be a real analytic representative of f such that (1)–(3) of Proposition 3.1 hold.

Lemma 3.2. If U is small enough, then (f )ˆ \ {0} is empty or a 1- dimensional manifold which has only finitely many topological components.

Proof. By (3), ifp∈(f )ˆ \{0}, thenpis a fold point, and the singular set is diffeomorphic to the real line in a neighbourhood of a fold point. Also,(f )ˆ \ {0}is a semianalytic set, and hence, its intersection with a small neighbourhood of 0 has only finitely many topological components.

LetD = {p ∈^R2 | p ≤ }and letS = {p∈ ^R2 | p = } =∂D . DefineS˜ (f )ˆ = ˆf⁻¹(S )andD˜ (f )ˆ = ˆf⁻¹(D ).

Lemma3.3. IfU is small enough, thenfˆSδfor small enoughδ >0.

Proof. By Lemma 3.2 there are only finitely many branches of(f )ˆ \{0}. By the Curve Selection Lemma, for each component B_i of (f )ˆ \ {0}we may choose an analytic curve γi : [0, ) → ^R2 such that γ (0) = 0 and γ_i(0, ) ⊂ B_i. The curvesfˆ◦γ_i are analytic and by (1) of Proposition 3.1, (fˆ◦γ_i)(t ) = 0 whent > 0 and therefore(fˆ◦γ_i)S_δi for smallδ_i > 0. If δ <miniδ_i, thenfˆ|(f )ˆ S_δ. This proves the lemma, sincefˆS_δat any regular point offˆbecause the dimensions of source and target are equal.

The proof of Lemma 3.3 actually gives us more information. Let(f )ˆ = f ((ˆ f )).ˆ

Corollary 3.4. IfU is small enough, then (f )ˆ \ {0} is empty or a 1-dimensional smooth manifold such that(f )ˆ Sδ for smallδ.

Letθ :R²→^Rbe given byθ (p)= p².

Lemma3.5. Ifδis small enough, then∇(θ◦ ˆf )(p) =0for allp∈Dδ\{0}. Proof. Iffˆ=

f1

f2

, thenθ◦ ˆf =f₁²+f₂². We compute

∇(θ ◦ ˆf )(p)=2

f₁∂f₁

∂x +f₂∂f₂

∂x, f₁∂f₁

∂y +f₂∂f₂

∂y

(p)

=2( f₁(p) f₂(p) )·Df (p)ˆ

Ifp /∈ (f ), thenˆ f (p)ˆ = 0 and Df (p)ˆ is invertible, and hence, ∇(θ ◦ f )(p)ˆ =0. Assume thatp ∈(f )ˆ andp = 0. By (1),f (p)ˆ =0 and by the above,

∇(θ◦ ˆf )(p)=0⇔ ˆf^T(p)Df (p)ˆ =0

⇔ ˆf (p)⊥ImDf (p)ˆ

⇔Df (p)(ˆ R²)+^R −f₂(p) f1(p)

=^R2.

Note that_−f

2(p) f1(p)

is a tangent vector atf (p)ˆ to the circleS _{ˆf (p)}. It therefore follows from Lemma 3.3 thatDf (p)(ˆ R²)+^R₋

f2(p) f1(p)

=^R2. This proves the lemma.

Lemma3.6 (Lojasiewicz). There is aρ >0and constantsC, r >0such that forp∈D_ρ, ˆf (p) ≥Cp^r.

Proof. Remember that 0 is an isolated zero offˆand apply IV 4.1 of [5].

Lemma3.7. For small >0,S˜ (f )ˆ is a compact 1-manifold diffeomorphic toS¹and0is in the bounded component ofR²\ ˜S (f ).ˆ

Proof. Letρ >0 be such that ˆf (p) ≥ Cp^r for allp ∈D_ρ. Such a ρexists by Lemma 3.6. If ≤Cρ^r, thenS˜ ⊂D_ρis closed and bounded, i.e.

compact. By Lemma 3.3, ifρ is small enough, thenf S_Cρ^r in which case

˜

S is a 1-dimensional smooth manifold.

Every component ofS˜ is diffeomorphic toS¹by the classification of smooth compact 1-manifolds. LetC be one such component. Then C is an equipo- tensial curve ofθ ◦ ˆf. If 0 is not in the bounded component ofR²\C, then θ◦ ˆf has an extremal pointpin the bounded component ofR²\C, and hence,

∇(θ◦ ˆf )(p)=0. According to Lemma 3.5, this is not possible for smallρ. It follows that 0 is in the bounded component ofR²\C.

Assume thatC andDare different components ofS˜ (f ). Then there areˆ two bounded components ofR²\(C∪D), one of them containing 0. The other component must contain an extremal point ofθ ◦ ˆf which is impossible for smallρ.

Figure 2 below illustrates some of the properties we have proven so far.

S˜

D˜

δ

f (f)

S

D_δ (f)

Figure2. Illustration of Lemma 3.2, Lemma 3.3, Corollary 3.4 and Lemma 3.7.

LetEδ= {p∈^R2| p< δ} =intDδand letE˜δ(f )ˆ = ˆf⁻¹(Eδ).

Lemma3.8. For smallδ >0the mapfˆ| ˜E_δ\ {0}:E_δ\ {0} →E_δ\ {0}is proper.

Proof. By Lemma 3.6 there areC, r, ρ >0 such that ˆf (p) ≥Cp^r for allp ∈D_ρ. Assume thatδis so small that max

δ,_δ

C

¹_r

< ρ. Redefine fˆ puttingfˆ := ˆf|E_ρ. Then D˜_δ(f )ˆ ⊂ D

(C^δ)^1r ⊂ E_ρ, and hence,D˜_δ(f )ˆ is compact.

LetK ⊂ E_δ \ {0}be a compact set. LetK˜ = (fˆ| ˜E_δ\ {0})⁻¹(K)and let (p_n)be a sequence inK. Then˜ (p_n)is a sequence inD˜_δ(f ), and hence, thereˆ is a subsequencep_n(k) ofp_nand a pointp ∈ ˜D_δ(f )ˆ such thatp_n(k) →pas k→ ∞. Thenf (pˆ n(k))→ ˆf (p)∈K, and hence,p∈ ˜K. It follows thatK˜ is compact and thatfˆ| ˜E_δ\ {0}is proper.

Proposition3.9. For small >0, the restrictionfˆ| ˜S (f )ˆ :S˜ (f )ˆ →S is stable.

Proof. It is enough to show thatfˆ| ˜S (f )ˆ has only Morse singularities and no singular double points. Corollary 3.4 implies thatS˜ (f )ˆ (f )ˆ close to the origin. We also observe that(fˆ| ˜S (f ))ˆ ⊂(f ). In fact,ˆ (fˆ| ˜S (f ))ˆ = (f )ˆ ∩ ˜S (f ). Letˆ p∈(fˆ| ˜S (f )), and letˆ βbe a centered chart aboutpin

˜

S (f ), and letˆ πbe the projection ofR²onto the lineLperpendicular to(f )ˆ atf (p). The restriction ofˆ πto a neighbourhood off (p)ˆ inS is a chart about f (p)ˆ inS . Let andbe diffeomorphisms of neighbourhoods ofp,f (p)ˆ inU,R²respectively such thatfˆ=◦F◦whereF (x, y)=(x, y²). Such diffeomorphisms exist becausepis a fold point offˆ. Now, choose a linear isomorphismT :L→^Rwhich identifiesLwithRsuch thatT (π(f (p)))ˆ =0.

Letα = (α1, α2) = ◦β⁻¹and letA= T ◦π ◦. Thenfˆ| ˜S (f )ˆ ∼A

A◦F◦α. Now we compute

(A◦F ◦α)(t )=A_xα₁(t )+2Ayα₂(t )α₂(t ) and

(A◦F ◦α)(t )=[Axxα₁(t )+2Axyα2(t )α₂(t )]α₁(t )+Axα₁(t ) +[Ayxα₁(t )+2Ayyα2(t )α₂(t )]·2α2(t )α₂(t ) +Ay[2(α₂(t ))²+2α2(t )α₂(t )].

Here all the partial derivatives ofAare to be taken atF ◦α(t ). Since there is no neighbourhood ofpinS˜ (f )ˆ restricted to whichfˆ| ˜S(f )ˆ is injective and sinceS˜ (f )ˆ (f ), we see from the normal formˆ F of folds thatα₁(0)=0 and α₂(0) = 0. We have also chosen α2(0) = 0. The choice of L gives