Question: how to classify homology group in maple?

1. is module in algebraic geometry or group cohomology to classify homology group?

2. is final result like this? for example,

a dictionary  or table store

homology group 1, key=invariant = 1 hole in topological space , value = module 1, module 2 module 3 etc

homology group 2, key=invariant = 2 holes in topological space , value =module 4, module 5, module 6 etc

homology group 3, key=invariant = 3 holes in topological space , value = module 7, module 8.. etc

 ... etc

3. i find betti number can count holes, however how to know the form for

all combination?

i mean if it is module, how to change the module to get the full combination

http://www.singular.uni-kl.de/Manual/html/sing_159.htm

i find betti number's input is just ideal, but it is not module

if ideal is enough, why need module?

how to permuate module? and what do it classify and result in module?