A Short Note on the Thom-Boardman Symbols of Differentiable Maps
aa r X i v : . [ m a t h . A C ] J un A Short Note on the Thom-Boardman Symbols of Differentiable Maps
Yulan Wang, Jiayuan Lin and Maorong Ge
Abstract
It is well known that Thom-Boardman symbols are realized by non-increasingsequences of nonnegative integers. A natural question is whether the converse is also true. In thispaper we answer this question affirmatively, that is, for any non-increasing sequence of nonnegativeintegers, there is a map-germ with the prescribed sequence as its Thom-Boardman symbol.
Thom-Boardman symbols were first introduced by R. Thom and later generalized by J.M. Boardman to classify singularities of differentiable maps. They are realized by non-increasing sequences of nonnegative integers. Although Thom-Boardman symbols havebeen around for over 50 years, in general to compute those numbers is extremely difficult.Before J. Lin and J. Wethington [3] proved R. Varley’s conjecture on the Thom-Boardmansymbols of polynomial multiplication maps, there were only sporadic known results. J. Linand J. Wethington [3] provides infinitely many examples of map-germs with distinct Thom-Boardman symbols. Since Thom-Boardman symbol are given by non-increasing sequencesof nonnegative integers, a natural question is whether the converse is also true. In thispaper we answer this question affirmatively. We prove that for any given non-increasingsequence of nonnegative integers, there is a map-germ with the prescribed sequence as itsThom-Boardman symbol.For the reader’s convenience, let us briefly recall the definition of Thom-Boardmansymbols from [2].Let x , · · · , x m be local coordinates on a differential manifold M of dimension m . Denote A the local ring of germs of differentiable functions at a point x ∈ M . For any ideal B in A , the Jacobian extension , ∆ k B , is the ideal spanned by B and all the minors of order k of the Jacobian matrix ( ∂φ i /∂x j ), denoted δB , formed from partial derivatives of functions φ i in B . We say that ∆ i B is critical if ∆ i B = A but ∆ i − B = A (just ∆ B = A when i = 1). That is, the critical extension of B is B adjoined with the least order minors of theJacobian matrix of B for which the extension does not coincide with the whole algebra.Suppose that N is another differential manifold of dimension n and y , · · · , y n be localcoordinates on it. For a differential map F : M → N, F = ( f , f , · · · , f n ), we denote J theideal generated by f , · · · , f n in A . Then ∆ k J is spanned by J and all the minors of order k of the Jacobian matrix δJ = ( ∂f i /∂x j ).Now we shift the lower indices to upper indices of the critical extensions by the rule∆ i J = ∆ m − i +1 J . We repeat the process described above with the resulting ideals until wehave a sequence of critical extensions of J , J ⊆ ∆ i J ⊆ ∆ i ∆ i J ⊆ · · · ⊆ ∆ i k ∆ i k − · · · ∆ i J ⊆ · · · . i , i , · · · , i k , · · · ) is called the Thom-Boardman symbol of J , denote T B ( J ). The purpose of switching the indices is that doing so allows us to express T B ( J ) as follows: i = corank( J ) , i = corank(∆ i J ) , · · · , i k = corank(∆ i k − · · · ∆ i J ) , · · · where the rank of ideal is defined to be the maximal number of independent coordinatesfrom the ideal and the corank is the number of variables minus the rank.We also need the following construction from [3].Let M n be the set of monic complex polynomials in one variable of degree n . M n ∼ = C n by the map sending f ( x ) = x n + a n − x n − + · · · + a to the n -tuple ( a , a , · · · , a n − ) ∈ C n .If we take f ( x ) of degree n as above and g ( x ) = x r + b r − x r − + · · · + b of degree r , then the product h ( x ) = f ( x ) g ( x ) is a monic polynomial of the form h ( x ) = x n + r + c n + r − x n + r − + · · · + c , where the c j ’s are polynomials in the coefficients of f and g . Thisgives us maps µ n,r : C n × C r → C n + r defined by ( a , · · · , a n − , b , · · · , b r − ) → ( c n + r − , · · · , c ) . Assume n ≥ r and consider the Euclidean algorithm applied to n and r : n = q r + r , < r < rr = q r + r , < r < r ... r k − = q k +1 r k , < r k < r k − . Let I ( n, r ) be the tuple given by the Euclidean algorithm on n and r : I ( n, r ) = ( r, · · · , r, r , · · · , r , · · · , r k , · · · , r k , , · · · )where r is repeated q times, and r i is repeated q i +1 times.Let I ( µ n,r ) be the ideal in the algebra A of germs at origin generated by c j ’s in the map µ n,r : C n × C r → C n + r . Denote T B ( I ( µ n,r )) the Thom-Boardman symbol of this ideal,Robert Varley conjectured that T B ( I ( µ n,r )) = I ( n, r ) for any n ≥ r . In [3], J. Lin and J.Wethington confirmed Varley’s Conjecture. For the reader’s convenience, let us state theirresult here. Theorem 1.1.
T B ( I ( µ n,r )) = I ( n, r ) for any n ≥ r . Professor J. M. Boardman provides us the following example which has constant Thom-Boardman symbol.
Example 1.2.
The zero map-germ at origin F : C a → C b has Thom-Boardman symbol ( a, a, · · · , a, · · · ) . µ ∞ ,a the map-germ in Example 1 .
2. We have the following theorem.
Theorem 1.3.
Let ( i , · · · , i , i , · · · , i , · · · , i k , · · · , i k , · · · ) be a non-increasing sequenceof nonnegative integers, where i > i > · · · > i k ≥ for some k , i j repeats l j times foreach j < k and i k repeats infinitely many times. Denote µ as the Cartesian product of themap-germs of µ ( i − i ) l ,i − i × µ ( i − i )( l + l ) ,i − i × · · · × µ ( i k − − i k )( l + l + ··· + l k − ) ,i k − − i k × µ ∞ ,i k at origin. Then ( i , · · · , i , i , · · · , i , · · · , i k , · · · , i k , · · · ) is the Thom-Boardman symbol ofthe map-germ µ .Remark . Historically there were only sporadic known Thom-Boardman symbols forsome map-germs. J. Lin and J. Wethington [3] gave infinitely many examples of map-germswith distinct Thom-Boardman symbols. However, Theorem 1 . . .
1, we immediately obtain Theorem 1 . Acknowledgments
The authors thank Professor J. M. Boardman for providing usExample 1 .
2, which is even not well-known among experts. We are grateful for his gen-erosity to allow us to use his example in Theorem 1 .
3. Without it, Theorem 1 . . We first prove that Thom-Boardman symbols have additive property under Cartesian prod-uct.
Proposition 2.1.
Let F t : M t → N t , t = 1 , be two differentiable maps. Then the Thom-Boardman symbol T B ( F ) of F = ( F , F ) : M × M → N × N is equal to the sum of T B ( F ) and T B ( F ) , where F = ( F , F ) : M × M → N × N is the Cartesian productof F t : M t → N t , t = 1 , .Proof. Denote
T B ( F ) = ( s , s , · · · , s p , · · · ) and T B ( F t ) = ( s ( t )1 , s ( t )2 , · · · , s ( t ) p , · · · ) , t = 1 , s p = s (1) p + s (2) p for all integer p ≥ x ( t )1 , · · · , x ( t ) m t be local coordinates of origin on the differential manifolds M t , t = 1 , y ( t )1 , · · · , y ( t ) n t be local coordinates on the differential manifolds N t , t = 1 ,
2. De-note F t : M t → N t , F t = ( f ( t )1 , · · · , f ( t ) n t ) and J F t the ideal generated by f ( t )1 , · · · , f ( t ) n t in3 ( t ) , t = 1 ,
2, the local ring of germs of differentiable functions at origin in M t . Denote J F the ideal generated by f (1)1 , · · · , f (1) n ; f (2)1 , · · · , f (2) n in A , the local ring of germs of dif-ferentiable functions at origin in M × M . Note that in general A (1) × A (2) A and J F × J F J F . However, it does not affect our conclusion on the additive propertyof Thom-Boardman symbols. Using induction, we will prove that ∆ s p ∆ s p − · · · ∆ s J F =(∆ s (1) p ∆ s (1) p − · · · ∆ s (1)1 J F ; ∆ s (2) p ∆ s (2) p − · · · ∆ s (2)1 J F ) in A , which immediately implies the con-clusion in Proposition 2 . p = 1, it is easy to see that the Jacobian matrix δJ F = (cid:18) δJ F δJ F (cid:19) . (2.1)Therefore corank( J F ) = corank( J F ) + corank( J F ), that is, s = s (1)1 + s (2)1 .The critical extension of ∆ s J F is spanned by f (1)1 , · · · , f (1) n ; f (2)1 , · · · , f (2) n and all theminors of order m + m − s + 1 of the Jacobian matrix δJ F . Any minor of order m + m − s + 1 of the Jacobian matrix δJ F contains either at least m − s (1)1 + 1 rows from δJ F or m − s (2)1 + 1 rows from δJ F , so ∆ s J F ⊆ (∆ s (1)1 J F ; ∆ s (2)1 J F ) in A . It is easy to seethat the inclusion ∆ s J F ⊆ (∆ s (1)1 J F ; ∆ s (2)1 J F ) is actually an equality because any minor oforder m t − s ( t )1 + 1 of δJ F t , t = 1 , m + m − s + 1of δJ F by a unit in A .Suppose that ∆ s p ∆ s p − · · · ∆ s J F = (∆ s (1) p ∆ s (1) p − · · · ∆ s (1)1 J F ; ∆ s (2) p ∆ s (2) p − · · · ∆ s (2)1 J F ) in A for some p ≥
1. Because the generators of ∆ s (1) p ∆ s (1) p − · · · ∆ s (1)1 J F involve only x (1)1 , · · · , x (1) m and the generators of ∆ s (2) p ∆ s (2) p − · · · ∆ s (2)1 J F involve only x (2)1 , · · · , x (2) m , the Jacobian matrix δ ∆ s p ∆ s p − · · · ∆ s J F has the form δ ∆ s p ∆ s p − · · · ∆ s J F = δ ∆ s (1) p ∆ s (1) p − · · · ∆ s (1)1 J F δ ∆ s (2) p ∆ s (2) p − · · · ∆ s (2)1 J F ! (2.2)The same argument as the case p = 1 gives s p +1 = s (1) p +1 + s (2) p +1 and ∆ s p +1 ∆ s p · · · ∆ s J F = (∆ s (1) p +1 ∆ s (1) p · · · ∆ s (1)1 J F ; ∆ s (2) p +1 ∆ s (2) p · · · ∆ s (2)1 J F ) in A . This completes the proof of Propo-sition 2 . . . . Proof. (Proof of Theorem 1 . k to prove Theorem 1 . k = 0, the sequence in Theorem 1 . µ ∞ ,i .Suppose that Theorem 1 . k = p , p ≥
0. When k = p + 1, by inductiveassumption and Theorem 1 .
1, the map-germ4 µ = µ [( i − i p +1 ) − ( i − i p +1 )] l , [( i − i p +1 ) − ( i − i p +1 )] × µ [( i − i p +1 ) − ( i − i p +1 )]( l + l ) , [( i − i p +1 ) − ( i − i p +1 )] ×· · ·× µ [( i p − i p +1 ) − ( i p +1 − i p +1 )]( l + l + ··· + l p ) , [( i p − i p +1 ) − ( i p +1 − i p +1 )] = µ ( i − i ) l ,i − i × µ ( i − i )( l + l ) ,i − i × · · · × µ ( i p − i p +1 )( l + l + ··· + l p ) ,i p − i p +1 has Thom-Boardman symbol ( i − i p +1 , · · · , i − i p +1 , i − i p +1 , · · · , i − i p +1 , · · · , i p − i p +1 , · · · , i p − i p +1 , , · · · , , · · · ). Now let µ = ˜ µ × µ ∞ ,i p +1 . ByProposition 2 .
1, it has Thom-Boardman symbol ( i , · · · , i , i , · · · , i , · · · , i p +1 , · · · , i p +1 , · · · ). This completes the proof of Theorem 1 . Remark . As an easy consequence of Proposition 2 . .
1, we also obtainthat the map-germs µ kr,r and Cartesian product of r copies of µ k, have the same Thom-Boardman symbols. So Theorem 1 . µ k, ’s and µ ∞ , . Thosespecial map-germs form building blocks for map-germs with arbitrary Thom-Boardmansymbols. References [1] M. Adams, C. McCrory, T. Shifrin, R. Varley,
Invariants of Gauss Maps of ThetaDivisors , Proc. Sympos. Pure Math., Vol 54, pp 1-8, Amer. Math. Soc., Providence, RI,1993.[2] V.I. Arnol ′ d, A.N. Varchenko, S.M. Gusein-Zade, Singularities of Differentiable Maps ,Volume I, Birh¨auser, Boston, 1985.[3] J. Lin, J. Wethington,
On the Thom-Boardman Symbols for Polynomial MultiplicationMaps, arXiv:0902.1518, submitted.
DEPARTMENT OF MATHEMATICS, SUNY CANTON, 34 CORNELL DRIVE, CANTON,NY 13617, USAANHUI ECONOMIC MANAGEMENT INSTITUTE, HEFEI, ANHUI, 230059, CHINA
E-mail address: [email protected]
DEPARTMENT OF MATHEMATICS, SUNY CANTON, 34 CORNELL DRIVE, CANTON,NY 13617, USA
E-mail address: [email protected]
DEPARTMENT OF MATHEMATICS, ANHUI UNIVERSITY, HEFEI, ANHUI, 230039, CHINA