# chi2 -- Euler characteristic of the 2nd Adams operation applied to a complex

## Synopsis

• Usage:
m = chi2 F
• Inputs:
• F, ,
• Outputs:

## Description

The definition:

chi2 F := eulerCharacteristic sym2 F - eulerCharacteristic wedge2 F.

Walker's proof that the sum of the Betti numbers is at least 2^{codim M), illustrated:

 i1 : S = ZZ/101[a,b,c] o1 = S o1 : PolynomialRing i2 : mm = ideal vars S o2 = ideal (a, b, c) o2 : Ideal of S i3 : M = S^1/(mm^2) o3 = cokernel | a2 ab ac b2 bc c2 | 1 o3 : S-module, quotient of S i4 : F = res M 1 6 8 3 o4 = S <-- S <-- S <-- S <-- 0 0 1 2 3 4 o4 : ChainComplex i5 : sumBetti = sum(4,i->rank F_i) o5 = 18 i6 : sumTor = sum(4,i->length(Tor_i(M,M))) o6 = 50 i7 : chi2 F == eulerCharacteristic sym2 F-eulerCharacteristic wedge2 F o7 = true i8 : 2^(codim M)*(length M) == chi2 F o8 = false i9 : chi2 F <= sumTor o9 = true i10 : sumTor <= sumBetti*(length M) o10 = true

## Caveat

Returns an error if any homology has infinite length

## Ways to use chi2 :

• "chi2(ChainComplex)"

## For the programmer

The object chi2 is .