CoCalc Logo Icon
StoreFeaturesDocsShareSupportNewsAboutSign UpSign In

Real-time collaboration for Jupyter Notebooks, Linux Terminals, LaTeX, VS Code, R IDE, and more,
all in one place.

| Download

GAP 4.8.9 installation with standard packages -- copy to your CoCalc project to get it

Views: 418346
Tweet
Share
Share
1
<Chapter><Heading> Cat-1-groups</Heading>

2


3
<Table Align="|l|" >

4


5
<Row>

6
<Item>

7
<Index>AutomorphismGroupAsCatOneGroup</Index>

8
<C>AutomorphismGroupAsCatOneGroup(G)</C>

9


10


11
<P/>

12
Inputs a group <M>G</M> and returns the Cat-1-group <M>C</M>

13
 corresponding to the crossed module <M>G\rightarrow Aut(G)</M>.

14
</Item>

15
</Row>

16


17
<Row>

18
<Item>

19
<Index>HomotopyGroup</Index>

20
<C>HomotopyGroup(C,n)</C>

21


22


23
<P/>

24
Inputs a cat-1-group  <M>C</M> and an integer n. It returns the <M>n</M>th homotopy group of <M>C</M>. 

25
 

26
</Item>

27
</Row>

28


29
<Row>

30
<Item>

31
<Index>HomotopyModule</Index>

32
<C>HomotopyModule(C,2)</C>

33


34


35
<P/>

36
Inputs a cat-1-group  <M>C</M> and an integer n=2. It returns the second

37
 homotopy group of <M>C</M> as a G-module (i.e. abelian G-outer group)

38
where G is the fundamental group of C.

39


40
</Item>

41
</Row>

42


43
<Row>

44
<Item>

45
<Index>QuasiIsomorph</Index>

46
<C>QuasiIsomorph(C)</C>

47


48


49
<P/>

50
Inputs a cat-1-group  <M>C</M> and  returns a cat-1-group <M>D</M> for which there exists some homomorphism <M>C\rightarrow D</M> that

51
 induces isomorphisms on homotopy groups.  

52


53
<P/> This function was implemented by <B>Le Van Luyen</B>. 

54


55
</Item>

56
</Row>

57


58


59
<Row>

60
<Item>

61
<Index>ModuleAsCatOneGroup</Index>

62
<C>ModuleAsCatOneGroup(G,alpha,M)</C>

63


64


65
<P/>

66
Inputs a group <M>G</M>, an abelian group <M>M</M> and

67
a homomorphism <M>\alpha\colon G\rightarrow Aut(M)</M>.

68
 It returns the Cat-1-group <M>C</M>

69
 corresponding th the zero crossed module <M>0\colon M\rightarrow G</M>.

70
</Item>

71
</Row>

72


73
<Row>

74
<Item>

75
<Index>MooreComplex</Index>

76
<C>MooreComplex(C)</C>

77


78


79
<P/>

80
Inputs a cat-1-group <M>C</M> and returns its Moore complex 

81
 as a 

82
G-complex (i.e. as a complex of groups considered as 1-outer groups).

83
</Item>

84
</Row>

85


86
<Row>

87
<Item>

88
<Index>NormalSubgroupAsCatOneGroup</Index>

89
<C>NormalSubgroupAsCatOneGroup(G,N)</C>

90


91


92
<P/>

93
Inputs a group <M>G</M> with normal subgroup <M>N</M>.

94
 It returns the Cat-1-group <M>C</M>

95
 corresponding th the inclusion

96
 crossed module <M> N\rightarrow G</M>.

97


98
</Item>

99
</Row>

100


101
<Row>

102
<Item>

103
<Index>XmodToHAP</Index>

104
<C>XmodToHAP(C)</C>

105


106


107
<P/>

108
Inputs a cat-1-group <M>C</M> 

109
 obtained from the Xmod package and

110
returns a cat-1-group <M>D</M> for which IsHapCatOneGroup(D) returns true.

111
 

112
<P/>

113
It returns "fail" id  <M>C</M> has not been produced by the Xmod package.  

114
</Item>

115
</Row>

116


117


118
</Table>

119
</Chapter>

120


121


122


123