PHYSICS: BARGMANNAND WIGNER
pairing of G to G' is defined and the products gl g2 (g1, g92 G) all have order two. The group Hn(wr, n; G) is isomorphic with the group of homomorphisms of H5(ir,,n) into G, and H,(r, n) is isomorphic with 7r. Hence Hn(7r, n; 7r) is isomorphic with the group of endomorphisms of 7r; let dn be the element of Hn(7r, n; 7r) which acts as the identity endomorphism of 7r. For n > 2, 70:7rn(X) -rw+1(X) is a homomorphism with values in 2(7rn+l(X)) and therefore a commutative self-pairing of 7rn(X) to 7r,+l(X) can be found such that a. a = 710(a). It then follows that a! (3 always has order two and therefore Sqn_2:Hn(7rn(X), n; 7rn(X)) -H+2(n(X) n; 7rn+1(X)) is defined and is independent of the particular choice of the self-pairing. Using this pairing, we find THEOREM 6. If n > 2, Sqn52(dn) = kn+2. 1 Freudenthal, H., Comp. Math., 5, 299-314 (1937). 2 Fox, R. H., Bull. Am. Math. Soc., 49, abstract 172 (1943). 3 Hurewicz, W., Proc. Akad. Amsterdam, 39, 117-126'(1936). 4 Hopf, H., Comm. Math. Helv., 17, 307-326 (1945). 6 These results depend on Pontrjagin's result that TWn+2(S) = 0 for n 2 3. A proof was outlined in C. R. Acad. Sci. U.R.S.S., 19, 361-363 (1938), but a complete proof has not yet appeared. 6 Eilenberg, S., and MacLane, S., Ann. Math., 46, 489-509 (1945). 7 Cf. ]ilenberg, S., and MacLane, S., Proc. Nat. Acad. Sci., 32, 277-280 (1946) for the case n = 1. 8 Eilenberg, S., and MacLane, S., Ann. Math., 43, 757-831 (1942). 9 Steenrod, N. E., Ibid., 48, 290-320 (1947).
STEREOSCOPIC ACUITY FOR VARIOUS LEVELS OF ILL UMI-
7Dirac, P. A. M., Proc. Roy. Soc., A, 155, 447-459 (1936). 8 The literature on relativistic wave equations is very extensive. Besides the papers quoted in reference 11, we only mention the book by de Broglie, L., Theorie gene'rale des particides a' spin (Paris, 1943), and the following articles which give a systematic account of the subject: Pauli, W., Rev. Mod. Phys., 13, 203-232 (1941); Bhabha, H. J., Rev. Mlod. Phys., 17, 203-209 (1945); Kramers, H. A., Belinfante, F. J., and Lubanski, J. K., Physica, VIII, 597-627 (1941). In this paper, the sum of (14) over all v was postulated; (14a) then has to be added as an independent equation (except for N = 1). Reference 11 uses these equations In the form given by Kramers, Belinfante and Lubanski. 9 One may derive this result in a more elegant way, without specializing the co6rdinate system. For the sake of brevity, we omit this derivation. 10 de Wet, J. S., Phys. Rev., 58, 236-242 (1940), in particular, p. 242. "Wigner, E. P., Z. Physik, (1947).
Net Spaces in Categorical Topology a
In the paper we introduce the notion of S‐nets, S a nonempty construct, as a generalization of the usual nets. S‐net spaces and continuous maps between them are then defined in a natural way. For S‐net spaces we introduce a number of axioms and study the categories obtained. We find some topological and Cartesian closed categories of S‐net spaces and describe their relations to certain known topological categories.