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**HapTutorial
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Chapter 1.
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Chapter 2.
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Chapter 3.
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\OML/ztmcm/m/it/10.95 H[]\OT1/ztmcm/m/n/10.95 ([]\OML/ztmcm/m/it/10.95 ; \U/msb
/m/n/10.95 Z\OT1/ztmcm/m/n/10.95 ) = 0$ \T1/ptm/m/n/10.95 since $[]$ is a $\OT1
/ztmcm/m/n/10.95 2$\T1/cmtt/m/n/10.95 -\T1/ptm/m/n/10.95 dimensional CW\T1/cmtt
/m/n/10.95 -\T1/ptm/m/n/10.95 space. The ex-act se-quence im-plies $\OML/ztmcm/
m/it/10.95 ^^Y[][] []
 []

[41] [42] [43] [44]
Chapter 4.
[45

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\T1/ptm/m/n/10.95 In 1935 K. Rei-de-meis-ter [[]][] clas-si-fied lens spaces up
 to ori-en-ta-tion pre-serv-ing
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[49] [50] [51]

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[54] [55] [56] [57] [58] [59] [60] [61] [62] [63]
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[]\T1/ptm/m/n/10.95 The fol-low-ing com-mands show that there are 367 com-bi-na
-to-ri-ally dif-fer-ent reg-u-lar
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[64]
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\T1/ptm/m/n/10.95 topes. A poly-tope of par-tic-u-lar in-ter-est, and one that 
ap-pears sev-eral times in the clas-
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Chapter 5.
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Chapter 6.

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[83] [84]
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[]\T1/ptm/m/n/10.95 Consider the group $\OML/ztmcm/m/it/10.95 H \OT1/ztmcm/m/n/
10.95 = \OML/ztmcm/m/it/10.95 SmallGroup\OT1/ztmcm/m/n/10.95 (64\OML/ztmcm/m/it
/10.95 ; \OT1/ztmcm/m/n/10.95 134)$\T1/ptm/m/n/10.95 . Con-sider the nor-mal su
b-group $\OML/ztmcm/m/it/10.95 N \OT1/ztmcm/m/n/10.95 =
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[85] [86] [87] [88] [89] [90]

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Chapter 7.
[94

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\T1/ptm/m/n/10.95 Suppose given a group ho-mo-mor-phism $\OML/ztmcm/m/it/10.95 
f\OT1/ztmcm/m/n/10.95 : \OML/ztmcm/m/it/10.95 G[] \OMS/ztmcm/m/n/10.95 ! \OML/z
tmcm/m/it/10.95 G[]$ \T1/ptm/m/n/10.95 and a $\OML/ztmcm/m/it/10.95 G[]$\T1/cmt
t/m/n/10.95 -\T1/ptm/m/n/10.95 module $\OML/ztmcm/m/it/10.95 A$\T1/ptm/m/n/10.9
5 . Then $\OML/ztmcm/m/it/10.95 A$ \T1/ptm/m/n/10.95 is nat-
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[106]
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\T1/ptm/m/n/10.95 In the oc-ta-he-dral case with $\OML/ztmcm/m/it/10.95 m \OT1/
ztmcm/m/n/10.95 = 1$ \T1/ptm/m/n/10.95 we ob-tain $$H\T1/cmtt/m/n/10.95 ^\\T1/p
tm/m/n/10.95 ast(\T1/cmtt/m/n/10.95 \\T1/ptm/m/n/10.95 Gamma,\T1/cmtt/m/n/10.95
 \\T1/ptm/m/n/10.95 mathbb Z) = \T1/cmtt/m/n/10.95 \\T1/ptm/m/n/10.95 mathbb
 []


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\T1/ptm/m/n/10.95 In the tetra-he-dral case with $\OML/ztmcm/m/it/10.95 m \OT1/
ztmcm/m/n/10.95 = 1$ \T1/ptm/m/n/10.95 we ob-tain $$H\T1/cmtt/m/n/10.95 ^\\T1/p
tm/m/n/10.95 ast(\T1/cmtt/m/n/10.95 \\T1/ptm/m/n/10.95 Gamma,\T1/cmtt/m/n/10.95
 \\T1/ptm/m/n/10.95 mathbb Z) = \T1/cmtt/m/n/10.95 \\T1/ptm/m/n/10.95 mathbb
 []

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\T1/ptm/m/n/10.95 A non\T1/cmtt/m/n/10.95 -\T1/ptm/m/n/10.95 standard $\OT1/ztm
cm/m/n/10.95 3$\T1/cmtt/m/n/10.95 -\T1/ptm/m/n/10.95 cocycle $\OML/ztmcm/m/it/1
0.95 f$ \T1/ptm/m/n/10.95 can be con-verted to a stan-dard one us-ing the com-m
and
 []

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[110] [111] [112] [113] [114] [115]
Chapter 8.

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\T1/ptm/m/n/10.95 com-putes a fi-nite di-men-sional graded ring equal to the co
-ho-mol-ogy ring $\OML/ztmcm/m/it/10.95 H[]\OT1/ztmcm/m/n/10.95 (\OML/ztmcm/m/i
t/10.95 G; \U/msb/m/n/10.95 Z[]\OT1/ztmcm/m/n/10.95 ) :=
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[123] [124] [125] [126]
Chapter 9.
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Chapter 10.
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[]\T1/ptm/m/n/10.95 The fol-low-ing com-mands con-struct the fil-tered chain co
m-plex un-der-ly-ing the Lyn-
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Chapter 11.
[136

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\T1/ptm/m/n/10.95 An im-ple-men-ta-tion of the above method for Bieber-bach gro
ups is also avail-able for ar-bi-
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\T1/ptm/m/n/10.95 trary crys-tal-lo-graphic groups. The fol-low-ing ex-am-ple c
on-structs a res-o-lu-tion for the group
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og-i-cal per-tur-ba-tion to con-struct
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\T1/ptm/m/n/10.95 a res-o-lu-tion for $\OML/ztmcm/m/it/10.95 G \OT1/ztmcm/m/n/1
0.95 = \OML/ztmcm/m/it/10.95 SL[]\OT1/ztmcm/m/n/10.95 (\OMS/ztmcm/m/n/10.95 O\O
T1/ztmcm/m/n/10.95 (\U/msb/m/n/10.95 Q\OT1/ztmcm/m/n/10.95 ([]))$\T1/ptm/m/n/10
.95 . The fi-nite com-plexes were con-tributed in-de-pen-
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\T1/ptm/m/n/10.95 dently by A. Rahm, M. Du-tour\T1/cmtt/m/n/10.95 -\T1/ptm/m/n/
10.95 Scikiric and S. Schoe-nen-beck and are stored in the folder
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\T1/ptm/m/n/10.95 res-o-lu-tion for $\OML/ztmcm/m/it/10.95 G \OT1/ztmcm/m/n/10.
95 = \OML/ztmcm/m/it/10.95 PSL[]\OT1/ztmcm/m/n/10.95 (\OMS/ztmcm/m/n/10.95 O\OT
1/ztmcm/m/n/10.95 (\U/msb/m/n/10.95 Q\OT1/ztmcm/m/n/10.95 ([]))$\T1/ptm/m/n/10.
95 . The fi-nite com-plexes were con-tributed in-de-pen-
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\T1/ptm/m/n/10.95 dently by A. Rahm, M. Du-tour\T1/cmtt/m/n/10.95 -\T1/ptm/m/n/
10.95 Scikiric and S. Schoe-nen-beck and are stored in the folder
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[][]\T1/ptm/b/n/14.4 Eilenberg\T1/cmtt/m/n/14.4 -\T1/ptm/b/n/14.4 MacLane space
s as sim-pli-cial groups (not rec-om-
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\T1/ptm/m/n/10.95 The fol-low-ing ex-am-ple con-cerns the Eilen-berg\T1/cmtt/m/
n/10.95 -\T1/ptm/m/n/10.95 MacLane space $\OML/ztmcm/m/it/10.95 X \OT1/ztmcm/m/
n/10.95 = \OML/ztmcm/m/it/10.95 K\OT1/ztmcm/m/n/10.95 (\U/msb/m/n/10.95 Z[]\OML
/ztmcm/m/it/10.95 ; \OT1/ztmcm/m/n/10.95 3)$ \T1/ptm/m/n/10.95 which is a
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[]\T1/ptm/m/n/10.95 The fol-low-ing com-mands use this func-tion to show that t
here are pre-cisely $\OT1/ztmcm/m/n/10.95 49$ \T1/ptm/m/n/10.95 dis-tinct
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Chapter 13.

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Chapter 14.
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Chapter 15.

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Chapter 16.

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