A Ash

From the second expression for Q, v^e see easily that it is positive definite, of arithmetic minimum 1, and v^ith twelve pairs of minimal vectors

±^2' ±^3. ±^4. ±(1^0 or 1,0 or 1,0 or 1), ±(2,1,1,1)

If î;^ ,i;j2 are these twelve minimal vectors, we check that i^/i^^, ,1^12 ^^12 ^P^^ the vector space of all symmetric 4x4 matrices Hence Q is the unique form with these minima and arithmetic minimum 1 In other words, {Q} is a 0-cell in X QED

Remark When n = 5, QXy) is semi-defimte and the proposition is false In fact, if t is a small positive number, the form

Ш= i yf-ii-t) I У.У,- (1+ ^] i yjs

1=1 l^i<j<4 \ ^/i 1

has arithmetic minimum 1 and for minimal vectors exactly ±e^, 5 ±^5^ ±(1,1,1,1,2)

The next proposition will have an important corollary that greatly simplifies computing the kernel of the boundary map

Say two cells a, a' oïX are of the same type if and only if there exists уеГ with

( T = Gy

Proposition 4.4. Fix b, 2^b^w—1, and assume 3^n^4 The cell cp^ in X is a codimens ion-one face of exactly b cells of type T2, two cells of type t^,, one cell of type т^,+ p and no other cells

Remark We can symbolize this assertion by writing

^со^ , = Ьт2 + 2т^, + т^,+ 1

Proof It IS obvious that if ЕсЖ" and Ае(т{Е) then М(А)эЕ The minimal vectors of o)jj are Mf,= {±e^, , ±e„, ±m^ and ±^1^+1} To find ^ш^,, we simply must relax the condition that +x be a minimal pair for x = e^, ,^„, m^, m^^^ ^ one at a time

If we omit ±ej^ with /c>fc-f 1, the remaining vectors don't spanlR", so there is no cell m X with exactly those minimal vectors

1 ) Omit ±^b+i Consider the lattice basis e^, ,ê^^-^, ,^„, m^^^ for Z" The hat indicates omission In terms of this basis, m^, = ^^ + ^2 + + e^, Hence if ye Г is the matrix that sends the standard basis to e^ ^^Ь' '^b+i? ^ы-2' >^«> we have УЩ = Щ Hence y~^M^- {±e^^^})= {±e^, , ±e„, ±mj and so ШьСт^,7

2 ) Omit ±e^ for 1 ^d^b Here consider the lattice basis e^, ,ê^, ,^„,wt^, In terms of this basis, 'Wb+i =^6+1 ^'^ь Hence if ^еГ sends the standard basis to ^b+i' ^b> ^1' '^d' ^b+i' ^n-i' ±^n (where the last sign is chosen so that dety = l) theny{e^+e2) = mj^^^ Thus у-\М^^-{±е^}) = {±е^, ,±e„, ±(^1 + 62)} and so ^b ^ '^27 There are Ь different possible values for d here

3 ) Omitm^ , ThenM^-{ + mJ = {±ei, , ±e„, ±mj^+^} md œ^CXf^+i 4) Omit m^+i Then М^,-{±т^^^^} = {±е^, ,±e„, ±mj and ш,Ст, QED