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Then F, G and F x G are convex sets. Denote (x1* . . X, wl Wpyl * * Yn, VI ... vq) by (.x, w-,y, v-) and for z E Rn let H. be the affine space {(x, w, y-, v): x+ y= z). Let J. be the affine space {(x-, wy, v): x = a}. For arbitrary z and a, F x G n Hn Ja is either empty or the Cartesian product of a p-dimensional open cube of volume f(a) with a q-dimensional open cube of volume g(z a. In either case the (p + q)dimensional volume of Hz n J4, nF x G is f(a)g(z--). Since Hz is the disjoint union of the Hi n J4 and the distance between flats H. n Ja and Hi n Ja is VX ii a'ij, the (p + q + n)-dimensional volume V(z) of F x G n Hz is given by

By showing that certain Pontrjagin classes are fibre cobordism invariants "of connected type" we produce infinitely many examples of fibre cobordism groups which are not finitely generated. 1. Introduction. If A is a closed manifold we shall denote by Nx the cobor- dism group of smooth fibrations over A with fibre p-dimensional closed manifolds. Here a cobordism between two such fibrations ci : Zi —* X and q2 : Z2 —► A, with fibres Mi and M2 respectively, is a smooth fibration q : W —* X with fibre a com- pact manifold N, such that dW — Zi H Z2, dN = Mi U M2 and the restriction of q to dW is qi H q2. In the usual manner cobordism classes form an abelian group under the disjoint union. The oriented version of these groups has been studied by M. Kreck (K), P. Melvin (M) and F. Bonahon (B) when A is a circle. A. Didierjean (Dl) computed the groups ^b*! Nx and showed that, if X is simply connected, any nonzero element of Nx can be represented by a fibration with fibre a projective plane RP2. In this note we shall only consider A to be a sphere of dimension 4n, and construct for some cobordism classes invariants of "connected type" arising from certain Pontrjagin classes. These invariants will allow us to deduce that some of these cobordism groups are not finitely generated and, in particular, to compute N2 . We wish to express our thanks to A. Didierjean for bringing the question of invariants of connected type to our attention (D2).

PROOF. Let r' be the result of deleting e, e', and X. The theorem holds for r' by induction. Any unicursal path on r' has the form 7172 ... 7r* where each 7ri is a path starting and ending at P but not meeting P between. Clearly n is the number of edges leaving P in r, and so is the same for all paths. Let X be the path ee' from P to P in r. We get all possible unicursal paths on r by starting with such paths on rI and inserting X, getting X7w1 * * * 7r., T.1X .. *nX . . . I 7ri * * * 7nAX. Assuming that e, e' are the last two edges in the chosen ordering of the edges, we have-(71 . . . 7-jiX-i+1 * * * 7rn) =E(7ri . . . 7) Thus E(7r) = (n+1) ,E(r') =0, the first sum being over all unicursal paths on r and the second over such paths on r'. We now consider Case 2 of [1]. If P =A, we can repeat the argument of Case 2 using B and C in place of B and A with only minor modifications. This will be possible provided C#A, but if C=A, the lemma applies with X = B. Suppose now that P = B. Let U be the set of unicursal paths on r starting at A, U' the set of such paths which begin with e, and Ui the set of unicursal paths on ri starting at A. Then the argument of [1, Case 2] shows that U= U'YUUi, a disjoint union. Since the theorem holds for U by what we have just proved, and also for Us, we see that E(r') = 0 where 7r' runs over all elements of U'. But there is a one-to-one correspondence between U' and the set of unicursal paths starting from B, given by ee'e, ... e. elel *.*.* ene. Since n =E-2 is even, e(ee'e1 . . . en) = -e(e' * * * e,e). Therefore the theorem holds in this case also. Finally, we consider Case 3. If there is an edge not meeting P, choose it for e4. Then P A, B and we are done. Suppose every edge meets P. Let P, A1, * * * I A. be the vertices. Then E=2V=2n+2.

The purpose of this note is to answer in the negative the following question raised by Gamelin [l]: if A A is a function algebra which has the property that X X , the spectrum of A A , is expressible as the union of two compact sets X 1 {X_1} and X 2 {X_2} which have as their intersection a peak point p p of X X , and if f ∈ ( C ( X ) f \in (C(X) satisfies f | x 1 ∈ A | x 1 f|{x_1} \in A|{x_1} and f | x 2 ∈ A | x 2 f|{x_2} \in A|{x_2} , then is f ∈ A f \in A ? The counterexample is obtained by the use of a construction which is applicable to general function algebras. Let A A be a function algebra and I I a proper closed ideal, denoting by A [ I ] A[I] the set { ( f , f + s ) : f ∈ A , s ∈ I } \{ (f,f + s):f \in A,s \in I\} , it is shown that A [ I ] A[I] is a function algebra which has as its spectrum two copies of the spectrum of A A identified along hull ( ( I ) (I) ).

Let A be a dendrite whose endpoints are dense and let A be the complement in A of a dense er-compact collection of endpoints of A. This paper investigates various general position properties that finite products of A and A possess. In particular, it is shown that (i) if X is an LCn-space that satisfies the disjoint n-cells property, then X X A satisfies the disjoint (n + l)-cells property, (ii) An X (—1,1) is a compact (n + l)-dimensional AR that satisfies the disjoint n-cells property, (iii) An+1 is a compact (n + l)-dimensional AR that satisfies the stronger general position property that maps of n-dimensional compacta into An+1 are approximable by both Z-raaps and Zn -embeddings, and (iv) An+1 is a topologically complete (n + l)-dimensional AR that satisfies the discrete n-cells property and as such, maps from topologically complete separable n-dimensional spaces into An+1 are strongly approximable by closed Zn-embeddings. 1. Introduction. It is well known that the standard (2n + l)-cell 72n+1 is a compact absolute retract that satisfies the disjoint n-cells property and, as such, admits an embedding of every compact n-dimensional metric space. Moreover, the collection of embeddings of an n-dimensional compactum X into J2"+1 forms a dense Gg in the function space C(X,I2n+1) (HW). Though there are compact n-dimensional spaces that satisfy the disjoint n-cells property and admit embed- dings of every n-dimensional compactum, there are no topologically complete n- dimensional absolute neighborhood retracts that satisfy this property, for any such space would contain uncountably many pairwise disjoint embedded n-cells and thereby violate a result of Sieklucki (Si). In this paper, we produce examples of compact (n + l)-dimensional absolute retracts that satisfy the disjoint n-cells property, and the collection of embeddings of an n-dimensional compactum X into any of these examples forms a dense Gs in the corresponding function space. The examples are particularly nice in that they arise as product spaces whose factors are compact 1-dimensional absolute retracts, equivalently, dendrites. The disjoint n-cells property is but one example of a variety of general posi- tion properties that have played an interesting and increasingly important role in Geometric Topology during the past fifteen years, particularly in the area of char- acterizations of manifolds modeled on various spaces. The disjoint disks property of

In 1967 Charles Fefferman proved a RadonNikodym theorem for finitely additive measures. We give an elementary proof of a generalization of this theorem. In this paper we give an elementary proof of the THEOREM. Let u, y be real-valued boundedfinitely additive set functions on an algebra E of subsets of the set S. Then for every e>0 there exists Ne and a u-simple function f on S such that EE2E for some Be>E we have vB>koc-oc. If Ee>E and EcB, then vE=vB-v(B\E)>koc-cc-kc=-cc; if EciE and E' S\B, then vE= v(B UE)-vB 0 such that a> -(6/2 and y> -(3/2, then there exist NeE and a qsimple function f on S such that Received by the editors August 27, 1970. AMS 1970 subject classifications. Primary 28-00, 28A25, 28A80. ? American Mathematical Society 1972 225 This content downloaded from 157.55.39.171 on Sat, 23 Jul 2016 04:06:02 UTC All use subject to http://about.jstor.org/terms 226 JAN PACHL [March Ee2E &E c Nl,uEj 2(yS+6) (,uS+(3)/12. Put Bk =B( k (by Lemma 1), \ 2(4uS+ (3) "2m/ m m A1=S\UBj, Ak=Bk_l\U Bj, k = 2, 3, * , m, j=l j=k (36(k -1) N =B9 f = > 2(4uS + ) 2m 29 thus 2 pS +(3/( (32 Its?(36 pE E and EcS \N, then E is the disjoint union of the sets E1, * * Em, where Ek==E flAk for k= 1, 2, , m. Moreover, since Ek C Ak C S\Bk, we have (3k (3 yEk2(S + ,)uEk >-2for k = 2,., m. 2(4uS + (3) 2m Hence 2 ( 2m)