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**HapTutorial
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[6]
Chapter 1.
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Chapter 2.
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Chapter 3.
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[]\T1/ptm/m/n/10.95 Continuing with the above ex-am-ple where $\OML/ztmcm/m/it/
10.95 Y$ \T1/ptm/m/n/10.95 is the real pro-jec-tive plane, we see that
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Chapter 4.
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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 PL-
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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 
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Chapter 5.
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Chapter 6.
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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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Chapter 7.
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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 $$H\T1/cmtt/m/n/10.95 ^\\T1/p
tm/m/n/10.95 ast(\T1/cmtt/m/n/10.95 \\T1/ptm/m/n/10.95 Gamma,\T1/cmtt/m/n/10.95
 \\T1/ptm/m/n/10.95 mathbb Z) = \T1/cmtt/m/n/10.95 \\T1/ptm/m/n/10.95 mathbb
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ztmcm/m/n/10.95 = 1$ \T1/ptm/m/n/10.95 we ob-tain $$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-standard $\OT1/ztmcm/m/n/10.95 3$\T1/ptm/m/n/10.95 -coc
ycle $\OML/ztmcm/m/it/10.95 f$ \T1/ptm/m/n/10.95 can be con-verted to a stan-da
rd one us-ing the com-mand
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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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[111]
Chapter 9.
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Chapter 10.
[114

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Chapter 11.
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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. Dutour-Scikiric and S. Schoe-nen-beck a
nd are stored in the folder
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og-i-cal per-tur-ba-tion to con-struct a
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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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nd are stored in the folder
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Chapter 12.
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[][]\T1/ptm/b/n/14.4 Eilenberg-MacLane spaces as sim-pli-cial groups (not rec-o
m-
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Chapter 13.

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\T1/ptm/m/n/10.95 can be used to com-pute res-o-lu-tions for groups whose data 
(pro-vided by Se-bas-
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\T1/ptm/m/n/10.95 tian Schoen-nen-beck, Alexan-der Rahm and Math-ieu Du-tour) i
s stored in the di-rec-tory
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[]\T1/ptm/m/n/10.95 The next com-mands first con-struct the con-gru-ence sub-gr
oup $\OT1/ztmcm/m/n/10.95 ^^@[](\OML/ztmcm/m/it/10.95 I\OT1/ztmcm/m/n/10.95 )$ 
\T1/ptm/m/n/10.95 of in-dex $\OT1/ztmcm/m/n/10.95 144$ \T1/ptm/m/n/10.95 in
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Chapter 14.
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Chapter 15.
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[]\T1/ptm/m/n/10.95 W. Thurston.  \T1/ptm/m/it/10.95 The Ge-om-e-try and Topol-
ogy of Three-Manifolds\T1/ptm/m/n/10.95 .
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