By Joseph V. Collins

Excerpt from An undemanding Exposition of Grassmann's Ausdehnungslehre, or thought of Extension

The sum qf any variety of vectors is located via becoming a member of the start element of the second one vector to the top element of the 1st, the start aspect of the 3rd to the tip aspect of the second one. etc; the vector from the start element of the 1st vector to the tip aspect of the final is the sum required.

The sum and distinction of 2 vectors are the diagonals of the parallelogram whose adjoining facets are the given vectors.

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**Example text**

S, let Ffj. be the generator corresponding to Hj. One has V ^ ' C VHi ; in case of a strict inequality denote by qj = Hks, where the product is over all indices 6 such that ms divides \Hj\ but doesn't divide \Hj\. (In order to have a unique notation, for SQ one has t = 0). 3. 1) S Q is an isomorphism except for the cases k — I -f 1, with I — 1 or 2. ,s. 2) If k < I —I, let IQ = (mi : . . 's of these numbers. F) = (m0r/l0)degE(F). Thus, E r is always one to one and S r is onto only if rriQr = IQ, with n > 0 if k = / — 1.

1 . Trivial invariant part, t h e case p > 1. T h e o r e m 3 . 1 . / / k = I -f 1 — 2/9, p > 1, iij = kjirij for j = 1 , . . , ra — p = n, rij multiple of mr , r = n + 1 , . . , ra, then i Jb+2m^ f^ ) 2Z P> 1 (nfci)mo/mm» P = 1 (0,ifn = 0 , r a = l ) . 2 that two extensions may differ by a multiple of (IIA:j)rao/m m , however the following explicit construction will be needed. Let (/o,<£o) the map D e an Y extension of (1,0) to [0,1] x BQ with norm 1 and of degree d. Consider S^EQUIVARIANT D E G R E E 43 fd(t,x0,z)=(^^{fo-^^---A^m3-ejznJmnt(i-t)(Ro-\^\),--^ + (1,0,0) where j runs from 1 to r?.

Once the extension to the ball C is performed, one extends for ip € [0, 27r]r by using the action of the group 7 , namely: f(t,x0,\zP\,z,lkz) = e2"'*/P/(iia:o,kpl,e-2"*/Pi,z) for*/|z| G A. 0i|^|e-«f-i^~ie-if-i^i) by the construction of / , thus / = e^/P/^^^j^i^e-^/P^^e-^/PS) is well defined. Furthermore = e l >/(t,a:o,2). D. Note that since lLp is the maximal group which leaves zp real and positive then this construction is compatible with the previous S -maps. Corollary 2 . 1 . If for all isotropy subgroups H of S then a S -map f : S IR /+1 \ {0} is trivial one has that dimV > W \ {0} has a non zero S -extension Thus, if k < I, n £ + 2 m ( S /+2n ) - 0.