Manifold (ConceptTopic, 4)

From Hi.gher. Space

(Difference between revisions)

Revision as of 02:00, 31 August 2008

A manifold is a shape formed from a regular base shape, where various edges are connected either with or without twists.

Square manifolds

These are the best known manifolds. There are eight of them shown as follows:

Group I0	Group I1		Group I2				Group S2
Square	Hose (uncapped cylinder)	Möbius strip	Torus	Klein figure 8	Klein bottle	Real projective plane	Sphere
http://teamikaria.com/dl/zebnuw8TVYvi5oa7PHJFVXjeWIzl6j5b_OqpV9YRZnQ1HR5q.png	http://teamikaria.com/dl/HGJdxDSgKdFFmJwMx3NmaeeTiQYuIdQTfVqyhFLWxV8c60WH.png	http://teamikaria.com/dl/iCJxkx0R_t4XE_yO1bF8QpBRH-XMi5nDn-ELIknrjyQDa9X8.png	http://teamikaria.com/dl/K0bGhfOG3hetb1QT2ev6gCwfdT_JioJJVxSOe65WDn2pyrQW.png	http://teamikaria.com/dl/LpJNtb-p34CMW2XuXAEHxPWu_Yo2bx-gXbyxr8eOQU8xdJAr.png	http://teamikaria.com/dl/GonihCjgciwA83T8wiya7qsB93p4mpXIi2_t-EWe11UwKv71.png	http://teamikaria.com/dl/4h4Ag96nOfc3GRzA8r_rwcqoGURpR8NFiLFWJ9VogyAGV2az.png	http://teamikaria.com/dl/6DfUWpRUCTvIcPzgXCfjaVzgQueSMNorAmgm8o87JArm5GhR.png
I	0	1	00	01	10	11	SS

To construct, first connect the red edges to each other, matching up the arrowheads, and then connect the blue arrows together in the same way. Edges without arrows are left unconnected.

The Klein figure 8 and Klein bottle are topologically equivalent, however they have been listed separately as they appear significantly different.

Cubic manifolds

There are 279 unique cubic manifolds out of 611 defined ones. Only sufficient examples and the most interesting are shown in the following table.

Group I0	Group I1	Group I2	Group I3		Group S2	Group S3
1 defined	8 defined	64 defined	512 defined		1 defined	26 defined
1 unique	6 unique	36 unique	216 unique		1 unique	20 unique
1 shown	2 shown	2 shown	3 shown		1 shown	4 shown
Cube	Dihose	Toric hose	Ditorus	Toric bottle	Spherical hose	Toraspherinder	Glome
	Möbial hose	Real projective planar hose	Real projective realm			Toraspherindric bottle	Toraspherindric dalma

@@ Line 6: / Line 6: @@
 {|
-!style="background-color: #EEE;"|Group 0
+!style="background-color: #EEE;"|Group I0
-!colspan="2" style="background-color: #EEE;"|Group 1
+!colspan="2" style="background-color: #EEE;"|Group I1
-!colspan="4" style="background-color: #EEE;"|Group 2
+!colspan="4" style="background-color: #EEE;"|Group I2
-!style="background-color: #EEE;"|Group S
+!style="background-color: #EEE;"|Group S2
 |-
 |width="12%" align="center"|[[Square]]
@@ Line 48: / Line 48: @@
 {|
-!style="background-color: #EEE;"|Group 0
+!style="background-color: #EEE;"|Group I0
-!style="background-color: #EEE;"|Group 1
+!style="background-color: #EEE;"|Group I1
-!style="background-color: #EEE;"|Group 2
+!style="background-color: #EEE;"|Group I2
-!style="background-color: #EEE;" colspan="2"|Group 3
+!style="background-color: #EEE;" colspan="2"|Group I3
-!style="background-color: #EEE;" colspan="3"|Group S
+!style="background-color: #EEE;"|Group S2
+!style="background-color: #EEE;" colspan="2"|Group S3
 |-
 |align="center"|1 defined
@@ Line 58: / Line 59: @@
 |align="center"|64 defined
 |align="center" colspan="2"|512 defined
-|align="center" colspan="3"|26 defined
+|align="center"|1 defined
+|align="center" colspan="2"|26 defined
 |-
 |align="center"|1 unique
@@ Line 64: / Line 66: @@
 |align="center"|36 unique
 |align="center" colspan="2"|216 unique
-|align="center" colspan="3"|20 unique
+|align="center"|1 unique
+|align="center" colspan="2"|20 unique
 |-
 |align="center"|1 shown
@@ Line 70: / Line 73: @@
 |align="center"|2 shown
 |align="center" colspan="2"|3 shown
-|align="center" colspan="3"|5 shown
+|align="center"|1 shown
+|align="center" colspan="2"|4 shown
 |-
 |align="center" width="12%"|[[Cube]]
@@ Line 77: / Line 81: @@
 |align="center" width="12%"|[[Ditorus]]
 |align="center" width="12%"|[[Toric bottle]]
+|align="center" width="12%"|[[Spherical hose]]
 |align="center" width="12%"|[[Toraspherinder]]
-|align="center" width="12%"|[[Toraspherindric bottle]]
+|align="center" width="12%"|[[Glome]]
-|align="center" width="12%"|[[Toraspherindric dalma]]
 |-
 |align="center"|
@@ Line 86: / Line 90: @@
 |align="center"|[[Real projective realm]]
 |align="center"|
-|align="center"|[[Spherical hose]]
-|align="center"|[[Glome]]
 |align="center"|
+|align="center"|[[Toraspherindric bottle]]
+|align="center"|[[Toraspherindric dalma]]
 |}
 [[Category:Topology]]