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 <td
style="text-align: center; vertical-align: top; color: rgb(0, 0, 102);"><big>About HAP: Enumerating homotopy 2-types 
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style="vertical-align: top; background-color: rgb(255, 255, 255); text-align: left;">A

 2-type is a CW-space X
whose
homotopy groups are trivial in dimensions n=0 and n>2. As explained
in a <a href="aboutCrossedMods.html">previous page</a> 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 
 
 <div style="text-align: center;">G --> K1 <-- K2
--> K3 <-- ... --> Kn <-- H 
 </div>
 
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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style="vertical-align: top; background-color: rgb(204, 255, 255);">THEOREM 
 
 <div
style="text-align: left; background-color: rgb(204, 255, 255);">The
2-types X and Y
are homotopy equivalent if and only if the associated cat1-groups
G
and
H
are
quasi-isomorphic. 
 </div>
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 <div style="text-align: center;"> 
 </div>
We mention that crossed modules
are a well-known alternative algebraic formulation of cat1-groups.
There
is
a
categorical
equivalence between these two algebraic notions.
We have opted to work through the language of cat1-groups. 
 </td>
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style="vertical-align: top; background-color: rgb(255, 255, 255);">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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style="background-color: rgb(255, 255, 204); vertical-align: top;">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 
 </td>
 </tr>
 <tr>
 <td
style="background-color: rgb(255, 255, 255); vertical-align: top;">The
following commands use the first and second homotopy groups to prove
that the list L contains at least 37 distinct quasi-isomorphism types. 
 </td>
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style="background-color: rgb(255, 255, 204); vertical-align: top;">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 
 </td>
 </tr>
 <tr>
 <td
style="vertical-align: top; background-color: rgb(255, 255, 255);">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.</td>
 </tr>
 <tr>
 <td
style="vertical-align: top; background-color: rgb(255, 255, 204);">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 
 </td>
 </tr>
 <tr>
 <td
style="vertical-align: top; background-color: rgb(255, 255, 255);">The
following commands show that the above list L
contains at most 51 distinct
quasi-isomorphism types.</td>
 </tr>
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 <td
style="vertical-align: top; background-color: rgb(255, 255, 204);">gap>
Q:=List(L,QuasiIsomorph);; 
gap> M:=[];; 
 
gap> for q in Q do 
> bool:=true;; 
> for m in M do 
> if not IsomorphismCatOneGroups(m,q)=fail then bool:=false; break;
fi; 
> od; 
> if bool then Add(M,q); fi; 
> od; 
 
gap> Length(M); 
51 
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 <td
style="vertical-align: top; background-color: rgb(255, 255, 255);"><big
style="font-weight: bold;">Identification of homotopy 2-types of low
order</big> 
 <div style="text-align: left;"> 
Let us define the "order" of a cat-1-group to be the order of its
underlying group. Le Van Luyen
has
incorporated
the
above techniques into a function <big style="font-weight: bold;">IdCatOneGroup(C)</big>
which inputs a cat-1-group C of "low order" and returns an integer pair
[n,k] that uniquely idenifies the quasi-isomorphism type of C. The
integer n is the order of a smallest cat-1-group quasi-isomorphic to C.
The integer k identifies a particular cat-1-group of order n. 
 
The following commands use this function to show that there are in fact
precisely 49 distinct quasi-isomorphism types of cat-1-groups of order
16. 
 </div>
 </td>
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 <td
style="vertical-align: top; background-color: rgb(255, 255, 204);">gap>
M:=List(L,IdCatOneGroup); 
[ [ 16, 1 ], [ 16, 2 ], [ 16, 3 ], [ 16, 4 ], [ 1, 1 ], [ 4, 4 ], [ 16,
6 ], [ 16, 7 ], [ 16, 8 ], [ 16, 5 ], [ 16, 9 ], [ 16, 10 ], [ 16, 5 ],
 
 [ 16, 11 ], [ 16, 12 ], [ 16, 13 ], [ 4, 1 ], [ 16, 14 ], [ 4, 2
], [ 16, 15 ], [ 16, 16 ], [ 16, 17 ], [ 16, 18 ], [ 16, 19 ], [ 16, 20
], 
 [ 16, 21 ], [ 16, 22 ], [ 16, 23 ], [ 16, 24 ], [ 4, 1 ], [ 4, 3
], [ 16, 25 ], [ 4, 4 ], [ 16, 26 ], [ 4, 4 ], [ 16, 27 ], [ 4, 2 ], [
4, 5 ], 
 [ 16, 28 ], [ 16, 29 ], [ 16, 30 ], [ 16, 31 ], [ 4, 4 ], [ 16,
32 ], [ 4, 4 ], [ 16, 33 ], [ 4, 5 ], [ 16, 34 ], [ 16, 35 ], [ 4, 5 ],
 
 [ 16, 36 ], [ 16, 37 ], [ 16, 38 ], [ 16, 39 ], [ 16, 40 ], [ 4,
3 ], [ 16, 41 ], [ 4, 4 ], [ 1, 1 ], [ 16, 42 ], [ 4, 5 ], [ 16, 43 ] ] 
gap> Length(SSortedList(M)); 
49 
 </td>
 </tr>
 <tr>
 <td
style="vertical-align: top; background-color: rgb(255, 255, 255);">The
following command identify the order and then quasi-isomorphism type of
the cat-1-group C associated to the crossed module 
 
 <div style="text-align: center;">G ---> Aut(G) 
 </div>
 
for G equal to the dihedral group of order 10. They then realize a
smallest possible cat-1-group D of this quasi-isomorphism type. (The
realization of the quasi-isomorphism type depends on the XMod package.) 
 </td>
 </tr>
 <tr>
 <td
style="vertical-align: top; background-color: rgb(255, 255, 204);">C:=AutomorphismGroupAsCatOneGroup(DihedralGroup(10)); 
Cat-1-group with underlying group Group( [ f1, f2, f3, f4, f5 ] ) . 
 
gap> Order(C); 
200 
gap> IdCatOneGroup(C); 
[ 2, 2 ] 
 
gap> D:=SmallCatOneGroup(2,2); 
Cat-1-group with underlying group C2 . 
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