Bott Periodicity and the Parallelizability of the spheres PDF

Volume of Prisms Cones

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Bott Periodicity and the Parallelizability of the spheres

Math. Soc. 65 (1959) 267-81. (2) ATIYAH

Transformation Groups of Spheres

subgroups of the rotation group of the n-dimensional sphere which we denote MONTGOMERY AND ZIPPIN Topological Transformation Groups I

Groups of Homotopy Spheres: I

The h-cobordism classes of homotopy n-spheres form an MILNOR J.

An invariant of plumbed homology spheres

A-spheres forms a group under connected sum which we denote ~A Math. 42(1977)

Enseignement scientifique

LES MATHÉMATIQUES DE. L'ENSEIGNEMENT SCIENTIFIQUE. LES CRISTAUX. Mots-clés. Sphère ; cube ; maille ; réseau ; volume ; cristaux. Références au programme.

Spheres and Cones

Spheres and Cones. Instructions. ?. Use black ink or ball-point pen. Surface area of sphere = 4ær² ... The sphere and the cube have the same volume.

Volume of Prisms Cones

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Corbettmaths

Question 1: Work out the surface area of each of these spheres. Give each answer to 2 decimal places (you may use a calculator). (a).

GCSE (1 – 9) Spheres and Cones Name:

The sphere and the cube have the same volume. Work out the radius of the sphere. Give your answer correct to 3 significant figures.

[ 223 ] BOT T

PERIODICIT

Y AN D TH E

PARALLELIZABILIT

Y OF TH E

SPHERE

S B

Y M. F. ATIYAH AND F. HIRZEBRUCH

Received

1 2 Apri l 196
0

Inti-eduction.

Th e theorem s o f Bot t (4) , (5) on the stable homotopy of the classical group s impl y tha t th e spher e S n i s no t parallelizabl e fo r n 4= 1,3,7. This was shown independentl y b y

Kervaire(8

an d

Milnor(7)

, (9). Another proof can be found in (3) , § 26-11. The work of J. F. Adams (on the non-existence of elements of Hopf invarian t one implie s mor e strongl y tha t S n wit h an y (perhap s extraordinary diffe r entiabl e structur e i s no t parallelizabl e i f n 4 = 1,3, 7. Thus there exist already four proof s fo r th e non-parallelizabilit y o f th e spheres th e first thre e mentione d relyin g o n th e Bot t theory a s give n i n (4), (5). The purpose of this note is to show how the refined for m o f Bott' s result s give n i n (6 lead s t o a ver y simpl e proo f o f th e non-paralleliz abilit y (onl y fo r th e usua l differentiabl e structure s o f th e spheres) W e shal l prov e i n fac t th e followin g theore m du e t o Milno r (9 ) which implies the non-parallelizability.

THEORE

M 1. There exists a

real vector bundle E, over the sphere 8 n with w n (£) =)= 0 only forn 1 2 4 or 8

Wiig) e Ji\Bg, Z

2 denote s th e it h

Stiefel-Whitne

y clas s o f th e rea l vecto r bundl e wit h bas e B v W e pu t w{£,) = £ w f (g)

Theore

1 is a consequence of

quotesdbs_dbs47.pdfusesText_47

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[PDF] Bott Periodicity and the Parallelizability of the spheres