therefore if one 6f them spirals down on A so does the other one. THEoREM 1. If two arcs with 0 and A as end-points do not cross each

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1 VOL. 39, 1953 MA THEMA TICS: R. L. MOORE 207 SPIRALS IN THE PLANE BY R. L. MooRi DEPARTMENT OF PURE MATHEMATICS, UNIVERSITY OF TEXAS Communicated December 22, 1952 In the plane, let J denote a definite circle, let 0 denote a definite point of J, let I denote the interior of J, let A denote a point of I and let a denote the straight line interval whose end-points are 0 and A. Let Z denote a definite point of a and LZ a definite arc lying in I and having only Z in common with a. Suppose a, is an arc from 0 to A lying wholly in 0 + I. From here on in this paper the word point will be restricted to mean point of 0 + I and, accordingly, all the arcs and other point sets under consideration will be understood to be subsets of Definition: The statement that the arc a, takes no step toward spiralling down on A means that it does not cross the straight line interval a. The statement that it takes one step toward spiralling down on A means that it crosses a. The statement that it takes two steps toward spiralling down on A means that there does not exist an arc from 0 to A which crosses neither ai nor a. If n is a positive integer, the statement that a1 takes n + 1 steps toward spiralling down on A means that there do not exist n arcs a2..., a.+1 from 0 to A such that if k < n + 1, ak+l does not cross ak, and a,,+n does not cross a. The statement that ai spirals down on A means that if n is a positive integer, a1 takes n steps toward spiralling down on A.1 It is clear that the following theorem holds true. THEoREM 1. If two arcs with 0 and A as end-points do not cross each other and one of them takes n steps toward spiralling down on A, then, if n is greater than 1, the other one takes n - 1 steps toward spiralling down on A and therefore if one 6f them spirals down on A so does the other one. It is to be observed that if an arc takes n + 1 steps toward spiralling down on a point then it takes n steps toward spiralling down on that point. Suppose BXC is an interval of the arc ai such that (1) C follows B in the order from 0 to A on a,, and (2) there exist a point B' preceding B and a point C' following C, in that order, such that the intervals B'BX and C'CX of a, both cross a, and BC is the only subinterval x of the interval B'C' of ai which has only its end-points in common with a and contains two arcs, one abutting on a from one side at one end-point of x and the other abutting on a from the other side at the other end-point of x. Then the segment BXC - (B + C) is called a winding of ai about A and this winding is called positive or negative according as it is or is not true that the interval BX of BXC and the arc LZ abut on a from the same side. No two windings of a1 about A have a point in common and if two of them have

2 208 MA THEMA TICS: R. L. MOORE PROC. N. A. S. a common end-point they are either both positive or both negative. It may be shown that if all the windings of a1 about A are positive or all of them are negative and the total number of them is the positive number n then cal takes n + 1 steps, and no more, toward spiralling down on A. Let (3 denote a finite or infinite simple sequence such that (1) each term of,b is a collection such that either every element of it is a positive winding of a, about A or every element of it is a negative winding of a, about A; (2) each winding of a1 about A is an element of one and only one term of 8; (3) if x and y are consecutive terms of (3 and each element of x is positive then each element of y is negative; and (4) if one collection follows another one in the sequence,b then every element of it follows every element of the other one in the order from 0 to A on the arc a1. Suppose no collection of the sequence (3 has infinitely many elements. Let yn denote a sequence bil,... whose nth term bln is the total number of windings in the nth term of (3 or minus that number according as every such winding is positive or every such winding is negative. Let 72' denote a sequence whose nth term is bln- 1 or - (IblnI - 1) according as bl,, is positive or negative. If every term of 72' is 0 then a, takes two steps, and no more, toward spiralling down on A. If there are infinitely many terms of 72', all of the same sign, such that no one of them is followed by any term of 72' with the opposite sign then a1 spirals down on A. Suppose that there do not exist infinitely many such terms of 72' and that some term of 72' is different from 0. Then, and only then, there exists a subsequence 72' Of 72' such that (1) every two consecutive terms of 72' have different signs; (2) if y is a term of 72' lying, in 72', between two consecutive terms x and z of 72' and having the same sign as x then there is no term of 72' lying, in 72', between x and y and having a sign opposite to that of x; and (3) every term of 72' different from 0 either is a term of 72' or lies, in 72', between some two consecutive terms of 72'. Let 72 denote a sequence such that (1) its nth term exists if and only if the nth term of 72' exists, and (2) 1f x is the nth term of 72' the nth term of 72 is X, plus the sum of all other terms y of 72' with the same sign as x, if there are any, such that y is not separated from x in 72' by any term of 72' with a sign opposite to that of x. Let 73' denote a sequence obtained from 72 as 72' was obtained from 7i. If every term of y3' is 0 then a, takes 3 steps, and no more, toward spiralling down on A. If there are infinitely many terms of 78', all of the same sign, such that no one of them is followed by any one of the opposite sign then a1 spirals down on A. If there are not infinitely many such terms of 73' and some term of 73' is different from 0 then, and only then, there exists a subsequence 73' of 73' satisfying with reference to 73' the conditions described above as being satisfied by 72' with reference to 72' and there exists a sequence 73 satisfying with reference to 78' and 73' the conditions described above as being satisfied by 72 with reference to 72' and 72'-

3 VOL. 39, 1953 MA THEMA TICS: R. L. MOORE 209 There exists a finite or infinite sequence yl, 'Y2', Y2", 72, 73', 73, 7s3, * * such that if y,z' exists then (1) if every one of its terms is 0, a, takes n steps, and no more toward spiralling down on A; (2) if there is an infinite subsequence of -y', with terms all of the same sign, such that no one of them is followed by any term of y.' of the opposite sign then a, spirals down on I A; and (3) if no such infinite subsequence of 7,1' exists and some term of Yn is not 0 then _yn", y. and y'+i exist and relations described above as holding between yl', 7yl, yi and 7Y2' continue to hold if they are replaced by 7Yn1 'Yn7, zn and yn+i, respectively. The following theorems hold true. THEOREM 2. If n is an integer greater than 1 the arc a1 takes n, and no more, steps toward spiralling down on A if, and only if, 7Yn' exists and each of its terms iso. THEOREM 3. The arc a, spirals down on A if, and only if, either (1) some collection of the sequence (3 has infinitely. many elements or (2) for some n, there is an infinite subsequence a of 7,n' such that all the terms of a have the same sign and no term of 7Yn' of the opposite sign follows any term of 6, or (3) the sequence yi, 72, 73,... 'is infinite. Definition: If the arc OB is a proper subset of the arc OC, the- arc OC is said to take n steps toward spiralling down on B if and only if OB does so and OC is said to spiral down on B if and only if OB does so. If the arc XY does not have 0 as an end-point the statement that it spirals down on the point Z means that it contains Z and is a subset of some arc which has 0 as an end-point and spirals down on Z. An arc is said to be a spiral if it spirals down on some point and a pointset that contains no spiral is said to be spiralfree. Definition: If the point C does not belong to the arc OB the arc OB is said to take n steps toward spiralling down on C if and only if every arc from 0 to C that contains OB takes n steps toward spiralling down on C. The following theorem holds true. THEOREM 4. If the point P is a limit point of the subsetm of the arc a1 and the arc a, does not take n steps toward spiralling down on any point of M then it does not take n steps toward spiralling down on P. THEOREM 5. If there are points on which the arc al spirals down the set of all points on which it spirals down is an inner limiting set and every i-nterval of a, contains uncountably many points on which a1 does not spiral down. Proof: Let K denote the set of all points on which ai does not spiral down. There exists a number m such that there is some point on which a1 does not take m steps toward spiralling down. For each positive integer n let K, denote the set of all points P of K such that a1 does not take m + n steps towards spiralling down on P. The point set K is the sum of the point sets K1, K2, K3,... and, by Theorem 3, each point set of this sequence is closed. It follows that a, - K is an inner limiting set.

4 210 MA THEMA TICS: R. L. MOORE PROC. N. A. S. Suppose X YZ is an interval of a1. If there is a straight interval containing both 0 and X YZ then a, does not spiral down on any point of X YZ. Suppose no straight interval contains both of them. Then there are two points E and F of X YZ such that 0, E and F are the vertices of a triangle OEF. For every point C on the side EF of this triangle let T(C) denote the first point of X YZ, in the order from 0, on the straight ray 0C. Unless T(C) is X or Z, T(C) is not a limit point of al - X YZ and therefore there is, on the straight interval from 0 to T(C), a point F such that the straight interval from T(C) to F contains no point of a, - X YZ and thus no p6int of a, except T(C) and therefore a, does not spiral down on T(C). Hence, in any case, there are uncountably many points of the interval X YZ on which a, does not spiral down. THEOREM 6. If M is a compact totally disconnected closed point set not containing 0 there exists an arc from 0 which spirals down on each point of M but on no point rot belonging to M. Proof for the Case Where M is Perfect: There exists a sequence G1, G2, G3,... such that (1) for each n, G. properly covers M and is a collection of 2" simple domains whose closures are mutually exclusive; (2) each domain and (3) M is the com- of Gn contains the closures of two domains of Gn+1; mon part of the point sets G1*, G2*, G3*...2 Let D1 and A2 denote the domains of G1 and let DiL and Di2 (i = 1, 2) denote the domains of G2 that lie in Di. For each sequence ji, J2.... in of numbers equal to 1 or 2, Djl... jn and DjJ.2.. *,2 denote the domains of Gn+1 that lie in the domain D 1j2.. jn of Gn. Let J0 denote a triangle 0B1B2 having 0 as one of its vertices and neither containing nor enclosing any point of Gi*. Let P1, P2, P3,... denote a sequence of points between B1 and B2, on the side B1B2 of this triangle, such that every point of B1B2 is a limit point of the set of all points of this sequence. Let 00 denote the point P1. Let bi and b2 denote the straight intervals B10 and B200, respectively. Let J1 and J2 denote two spiral free simple closed curves containing bi and b2, respectively, having only the point 00 in common, lying, except for b1 and b2, wholly in the exterior of Jo, containing no point of Gi*, but containing two arcs bl' and b2', respectively, such that (1) neither of them intersects bi or b2, and (2) if P is a point of D1 (i 1, 2) and t is an arc lying wholly in J1 plus its interior except that its end-points belong to b1 and b1', respectively, then t takes one step toward spiralling down on P. There exist two spiral-free simple closed curves J1' and J2' containing bi' and b2', respectively, lying, except for these arcs, wholly without Ji, J2 and each other and containing no point of G2* but such that J1' (i = 1, 2) contains an arc BjgBj2lying wholly in DLi such that if B1, Bl, Bi2 and 0, lie in that order on the boundary of the complementary domain of J1 + Jf' that contains 0, and 1' denotes the interior of J1', then!,'-di and Di - Di.i1' are both connected. Let Q. denote the

5 VOL. 39, 1953 MA THEMA TICS: R. L. MOORE 211 collection of all straight intervals from 0 to points of the interval B1B2. Let Qi (i = 1, 2) denote a collection of mutually exclusive spiral free arcs, filling up J1 plus Jf' plus their interiors 1i and Ii', such that if q is an arc of this collection its end-points are q-bi and q-(bibi2). Let O0 denote the point of Bi1Bi2 which is an end-point of the arc of the collection Qi whose other end-point is the first point of the sequence P1, P2,... which belongs to bi, let bil (i = 1, 2) denote the interval Bi1Oi of Bi1Bi2 and let bi2 denote the interval B12Oi of Bi2Bi2. There exist four spiral-free simple closed curves Jl, J12, J21 and J22 such that (1) no one of them intersects G2* or contains a point on or within any other one of them or of Jo, J1, J2, J1' or J2' except that Jij contains bcj, (2) Jij (i, j = 1, 2) lies wholly in Di and contains an arc bej', not intersecting bij, such that if P is a point of Dij and t is an arc lying wholly in Jj plus its interior except that its end-points belong to bj and bj', respectively, then t takes 2 steps toward spiralling down on P. There exist four spiral-free simple closed curves J.l', J12', J21' and J22' containing bf,', b12', b2l' and b22', respectively, lying except for these arcs wholly without J1, J12, J21,.22 and each other and containing no point of G* but such that Jij lies in Di and contains an arc Bij,Bi2, lying whony in Dij, such that B1j, Bijl, Bif2 and O0 lie in that order on theboundary of the complementary domain of Jij + Jij' that contains 0 and if Ii' denotes the interior of Jij' then Is>'.Dij and D,j - D1j-I,j' are both connected. Let Qt. (i, j = 1, 2) denote a collection of mutually exclusive spiral-free arcs, filling up Jij + ij + Jij' + Ij', such that if q is an arc of this collection its end-points are q-bij and q. (Bij,B112). For each point X of Bij1B112 let T(X) denote the point of B1B2 which is an end-point of the arc of the collection Qi whose other end-point is an end-point of an arc of the collection Qij whose other end-point is X. Let Oj denote the point of BijlBij2 which is an end-point of the arc of the collection Qij whose other end-point is an end-point of an arc of the collection Q, which has as its other end-point the point of lowest subscript in the sequence P1, P2,..., which lies on bi between T(B1j1) and T(Bi12). Continue this process indefinitely. Suppose k,, k2,... kn,, kn+1 is a sequence of numbers each equal to 1 or 2. The arc bklk,... k is the common part of the simple closed curves Jtk,k*. kn and Jkk,lk *.*.kn If Ikk2* *kn and 'kk... kn denote the interiors of Jkk,.. kn and Jklk... kn, respectively, the points.. Bklk2 * kn, Bkk... Bkk22 * * kn2and... k lie in that order on the... boundary ofjk,k2 kn + Jkn2... kn + bk1k,.. *knp. Qklk, * k is a collection of mutually exclusive spiral free arcs filling up Jkkk. kn + Ik,k2... kn + Jk2... kn + Ikk2 *.. kf such that if q is an arc of this collection its end-points are q- bkl2... kn and q, (Bklk2 *. kn,bktk2 * * * kn2). For each point X of the arc Bk,k,.. kn,bklk2... kn2y T(X) denotes a point of bkl for which there exists a sequence of arcs tl, t2,... t. such that for each i (1 < i < n) t1 is an arc of the collection Qk,... ki and every two consecutive arcs of this

6 212 MA THEMA TICS: R. L. MOORE PROC. N. A. S sequence have a common end-point and T(X) is an end-point of t1 and X is an end-point of t,. The point 0kjk *.. kn is a point of Bk,k2 * * * kn,bkk2... kn2 such that T(Ok1k,. kn) is the point of lowest subscript in the sequence P1, P2,... which lies on bk, between T(Bklk. * * knl) and T(Bklk * * * kn2). The intervals Bk1k, *. k,loki 2... kn and BkA2.... kn2oklk2.. kn are denoted by... bklk k.1 and b1k,2... kn2, respectively. The simple closed curves Jkk,..ḳ. and Jkk2... kn2 have only 0k,k... kn in common and they intersect Jklk2.. kn in bklk2 knl and bk1k2... kn2p respectively. The simple closed curve Jklkt... knkn+i lies wholly in the domain Dk12k... knand so does AJ,k2.. knk"+j and if P is a point of Dk1k2. - knkn+ every arc of the collection Qkk,* knkn+i takes n steps toward spiralling down on P. Let Z denote a collection of simple disks such that z belongs to Z if, and only if, either z is the triangle OB1B2 plus its interior or, for some finite sequence k,... kneach term of which is 1 or 2, Z is Jk,... kn + Ik, * * * kn or J1ki... kn + Itk. * * * k,- The sum of all the disks of the collection Z is a disk whose boundary J is a simple closed curve containing M. The arc 013B2 on J spirals down on every point of M and on no point that does not belong to,,. THEOREM'7. If 01B32 is a triangle there exists a collection G of arcs such that (1) each of them has 0 as one of its end-points and no two of them have any point in common except 0, (2) every straight interval with one end-point at 0 and the other one on the side B1B2 of this triangle is a subset of some arc of the collection G and each arc of the collection G contains one such interval, (3) each arc of the collection G spirals down on some point, and (4) the set of all points P such that some arc of G spirals down on P is a perfect point set. Proof: Using the notation of the above proof of a special case of Theorem 5, let Q denote a collection such that q is an element of it if and only if there exists a finite sequence ki,..., k,n of which each element is 1 or 2, such that q is an arc of the collection Qkk2... kn Suppose X is a point of the straight interval B1B2 not belonging to the sequence P1, P2.Then there is only one subcollection Q(X) of Q such that X plus the sum of the arcs of Q(X) plus some point of K is an arc from X to that point of K. Let T(X) denote that point of K. If X is a point of the sequence P1, P2,... there are only two such collections, Q1(X) and Q2(X). In this-case, for each number i which is equal either to 1 or to 2, let Tj(X) denote a point of K such that X + T(X) + Qj*(X) is an arc with X and Tj(X) as its end-points. There exists a sequence of mutually exclusive spiral-free arcs a,, a2,... such that, for each n, an has T1(Pn) and T2(P.) as its end-points and lies, except for those points, wholly in the exterior of J. Let G' denote a col lection such that g belongs to it if, and only if, either (1) g is X + Q*(X) + T(X) for some point X of B1B2 not belonging to P1, P2,..., or (2) for some n, g is Pn + Ql*(Pn) + an, Let G denote a collection of arcs such that the arc g belongs to it if, and only if, it is an arc of G' plus some straight interval

7 VOL. 39, MA THEMA TICS: N. E. STEENROD 213 with one end-point at 0. The collection G fulfills all the requirements of the conclusion of Theorem 6. 1 Professor Eilenberg has indicated to me that Waraszkiewicz has considered certain "spirals" on a cone which bear to one another a relationship similar to the relationship that ail bears to a according to this definition. Cf. Waraszkiewicz, S., Fundamenta Mathematicae, 22, (1934) and Ibid., 23, (1934). 2The notation G* is used to denote the sum of all the point sets of the collection G. HOMOLOGY GROUPS OF SYMMETRIC GROUPS AND REDUCED POWER OPERATIONS By N. E. STEENROD DEPARTMENT OF MATHEMATICS, PRINCETON UNIVERSITY Communicated by S. Lefschetz, January 13, Introduction.-We shall give a generalization of the reduced power operations' which connects them with the homology groups Hj(S.) of the symmetric group S. of degree n. Briefly if u is a q-dimensional cohomology class of a complex X, and c e H,(S.), then we define a cohomology class u"/c e H"n - i(x) called the nth power reduced by c. These operations include those defined previously. The broader viewpoint gives natural explanations of relations among cyclic reduced powers found by Adem2 and Thom.3 Their relations are homology relations in Sn. The new definition was inspired by the work of Adem. 2. Definition of Reduced Powers.-Let XI denote the product complex of n copies of the complex X. An element u of the group CQ(X; B), of q-cochains of X with coefficient group B, has an nth power defined by u". (oi X... X n = (u al) 0 *. 0* (u an) (2.1) where ois a cell ofx and uaoeb is the value of u on at. Thus un e C"(X"; B") where Bn is the tensor product of n copies of B. Let T be a permutation group of the factors of Xn. Each a e Tr gives an automorphism of the integral chain groups CQ(Xn). For example, if a interchanges the first two factors, then a(0f1 X 2 X. X an) = 1)'a2 X a1 X 3 X... X OS (2.2) where r, s are the dimensions of al, 0'2. We require u" to lpe an equivariant cochain, i.e., un. a(oax... Xo,,.) = a(un. a,x... Xc,). (2.3) This and 2.1, 2.2 determine the operation of a in Bn, namely, if q is even, a permutes the factors of Bn in the same manner as the factors of XI; and,