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A
2-type is a CW-space X
whose
homotopy groups are trivial in dimensions n=0 and n>2. As explained
in a previous page the homotopy
type of such a space can be captured algebraically by a cat1-group
G. Let us consider two 2-types X, Y represented by cat1-groups G, H. If X and Y are homotopy equivalent then there exists a sequence of morphisms of cat1-groups G --> K1 <-- K2
--> K3 <-- ... --> Kn <-- H
each morphism inducing an isomorphism on homotopy groups. When such a sequence of morphisms exists we say that G is quasi-isomorphic to H. We have the following result.
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All
small cat1-groups G have been listed up to isomorphism
in the GAP package XMod. For example, the following commands produce a
list L of all of the 62 non-isomorphic cat1-groups whose
underlying group has order 16. |
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gap>
LoadPackage("xmod"); gap> L:=[];; gap> for n in [1..NrSmallGroups(16)] do > k:=Cat1Select(16,n);; > for m in [1..k] do > G:=Cat1Select(16,n,m);; > Add(L,XmodToHAP(G)); > od;od; gap> Length(L); 62 |
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The
following commands use the first and second homotopy groups to prove
that the list L contains at least 37 distinct quasi-isomorphism types. |
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gap>
Invariants:=function(G) > local inv; > inv:=[]; > inv[1]:=IdGroup(HomotopyGroup(G,1)); > inv[2]:=IdGroup(HomotopyGroup(G,2)); > return inv; > end;; gap> C:=Classify(L,Invariants);; gap> Length(C); 37 |
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The following commands use second and third integral homology in conjunction with the first two homotopy groups to prove that the list L contains at least 49 distinct quasi-isomorphism types. | |||
gap>
Invariants:=function(G) > local inv; > inv:=[]; > inv[1]:=IdGroup(HomotopyGroup(G,1)); > inv[2]:=IdGroup(HomotopyGroup(G,2)); > inv[3]:=Homology(G,2); > inv[4]:=Homology(G,3); > return inv; > end;; gap> C:=Classify(L,Invariants);; gap> Length(C); 49 |
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Similar commands can be used to prove that there are exactly 6 distinct quasi-isomorphism types of cat1-group of order 4. There is obviously just one quasi-isomorphism types of cat1-groups of order 1. Hence the following commands show that the above list L contains at most 52 (=6+1+45) distinct quasi-isomorphism types. | |||
gap>
Q:=List(L,QuasiIsomorph);; gap> List(Q,Size); [ 16, 16, 16, 16, 16, 16, 1, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 4, 16, 4, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 4, 4, 16, 4, 16, 4, 16, 4, 4, 16, 16, 16, 16, 4, 16, 4, 16, 4, 16, 16, 4, 16, 16, 16, 16, 16, 4, 16, 4, 1, 16, 4 ] |
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