#Errorless mathematics pdf free download archive

The implications (1), (2), (3) represent cases of Theorem which are already known. Rn is a division algebra over the reals - n =2 Sn-1, with its usual differentiable n = 2 4 or 8 structure, is parallelisable ) (6) 2 Sn-1, with some (perhaps extraordinary) differentiable structure, is parallelisable s 0 t~(5) Sn-1 is an H-space There is an element of Hopf invariant one in 212n -(Sn) 11(1) j~i(2) (3) (4) n = 2 or 4r n= 2m n # 16 n = 2,4, or 8. 20ДЈ HOPF INVARIANT ONE 21 Rn is a normed algebra over the reals at = n = 2, 4, or 8. The remarks above, and certain other known theorems, may be summarized by the following diagram of implications. ) The case n = 1 is both trivial and exceptional, and we agree to exclude it from this point on. In fact, the systems of complex numbers, quaternions and Cayley numbers provide continuous products on the euclidean spaces R2, R4 and RI from these one obtains products on the unit spheres in these spaces, that is, on S1, S3 and S7. It is also classical that the conclusions are false for n = 2, 4 and 8. Various results in homotopy-theory have been shown to depend on the truth or falsity of these conclusions. Sqn: Hm(K Z2)- Hm+n(K Z2) It is a classical result that the four conclusions are equivalent. (d) Let K = Sm U Em+n be a CW-complex formed by attaching an (m + n)-cell Em+n to the m-sphere Sm. (b) In the homotopy group r2~n3(sn-1), the Whitehead product in w2rn1(sn). THEOREM Unless n = 1, 2, 4 or 8, we have the following conclusions: (a) The sphere Sn-1 is not an H-space. In particular, Hm(Y G) is the mth singular cohomology group of the space Y with coefficients in the group G. For the remaining notations, which are standard in homotopy-theory, we refer the reader to. An H-space is a space which admits a continuous product with unit. A continuous product with unit on a space X is a continuous map p: X x X -X with a point e of X such that p(x, e) = p(e, x) = x. It is the object of this paper to prove a theorem in homotopy-theory, which follows as Theorem In stating it, we use one definition. ADAMS (Received April 20, 1958) (Revised January 25, 1960) CHAPTER 1. 1, July, 1960 Printed in Japan ON THE NON-EXISTENCE OF ELEMENTS OF HOPF INVARIANT ONE BY J. For more information about JSTOR, please contact Annals of Mathematics is collaborating with JSTOR to digitize, preserve and extend access to The Annals of Mathematics.Дў ANNALS OF MIATHEMATICS Vol. We use information technology and tools to increase productivity and facilitate new forms of scholarship. JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive.

#Errorless mathematics pdf free download archive

1 (Jul., 1960), pp Published by: Annals of Mathematics Stable URL: Accessed: 28/04/ :48 Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at. Adams Reviewed work(s): Source: The Annals of Mathematics, Second Series, Vol. 1 Annals of Mathematics On the Non-Existence of Elements of Hopf Invariant One Author(s): J.