G2 geometry

Current hits: # 107345


0, 1
29, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 113, 115, 117, 119, 121, 122, 123, 125, 127, 129, 131, 133, 135, 137, 139, 141, 143, 145, 147, 149, 151, 153, 155, 157, 159, 161, 163, 165, 167, 169, 171, 173, 175, 177, 179, 181, 183, 187, 189, 195, 197, 239
(6, 1, 16, 13), (6, 1, 1, 0), (4, 8, 1, 0), (4, 5, 1, 0), (6, 1, 2, 1), (6, 1, 26, 3), (6, 2, 2, 1), (6, 2, 1, 0), (6, 1, 52, 29), (6, 1, 4, 1), (4, 4, 8, 7), (4, 4, 7, 6), (4, 1, 6, 5), (4, 1, 3, 2), (4, 1, 3, 1), (4, 1, 2, 1), (4, 2, 1, 0), (4, 2, 5, 2), (4, 4, 5, 1), (4, 4, 13, 3), (4, 4, 1, 0), (4, 3, 1, 0), (6, 2, 3, 1), (6, 3, 1, 0), (8, 8, 4, 3), (8, 8, 1, 0), (8, 4, 2, 1), (8, 4, 19, 14), [12], [24], [8], [6], [4], [2], (8, 4, 1, 0), (8, 2, 5, 4), (6, 6, 1, 0), (6, 4, 8, 5), (6, 4, 1, 0), (6, 3, 2, 1), (6, 8, 1, 0), (8, 1, 1, 0), (8, 2, 1, 0), (8, 16, 2, 1), (8, 1, 3, 1), (4, 1, 16, 15), (4, 1, 12, 11), (2, 1, 21, 4), (2, 1, 2, 1), (2, 1, 16, 5), (2, 1, 15, 4), (2, 1, 3, 1), (2, 1, 36, 19), (2, 1, 6, 1), (2, 1, 40, 37), (2, 1, 4, 3), (2, 1, 4, 1), (2, 1, 116, 107), (2, 1, 10, 7), (12, 2, 2, 1), (12, 2, 1, 0), (12, 16, 3, 2), (12, 16, 1, 0), (12, 2, 4, 3), (12, 3, 1, 0), (2, 1, 1, 0), (12, 4, 1, 0), (12, 3, 4, 3), (12, 3, 2, 1), (2, 1, 8, 1), (2, 1, 8, 5), (2, 4, 3, 1), (2, 4, 13, 8), (2, 4, 1, 0), (2, 30, 1, 0), (2, 4, 5, 4), (2, 4, 8, 7), (4, 1, 1, 0), (2, 8, 1, 0), (2, 6, 1, 0), (2, 5, 1, 0), (2, 3, 1, 0), (2, 20, 5, 3), (2, 16, 1, 0), (2, 11, 1, 0), (2, 10, 1, 0), (2, 1, 9, 4), (2, 18, 1, 0), (2, 2, 1, 0), (2, 20, 1, 0), (2, 2, 2, 1), (2, 2, 165, 152), (12, 1, 1, 0)
Fail, (1, 2), (0, 2), (0, 1)
(48, 72), (36, 72), (30, 36), (6, 12), (6, 24), (66, 72), (60, 72), (6, 36), (24, 72), (24, 36), (0, 72), (0, 36), (0, 24), (12, 24), (12, 36), (18, 24), (12, 72), (0, 12)

Explanitory Note

This site is intended to display data relevant to G2 manifolds constructed via the twisted connected sum method. It is not entirely clear who, beyond the author and the author's supervisor, will find themselves trying to navigate around this utility since it is a little niche. As such, little explaination is given here. If you think this might be interesting to you, but you've no idea what you're looking at then message me dw580 at bath.ac.uk

Terminology, naming conventions, and choices of bases or markings roughly follow those found in this paper and this paper. In particular, tcs refers to twisted connected sums, bbs refers to building blocks, and prebbs refers to things from which we derive building blocks such as fanos 3-folds.

Warnings:
  • It surely true that some of the input data is wrong through typos, bugs and/or bad maths. These are corrected as and when they are found.
  • In the case of a skew matching, further checks on the genericity conditions are needed to ensure the existence of G2 metrics. For some classes of building blocks these can be checked by a machine but this has yet to be added here. Thus not all skew matchings presented will be genuine G2 manifolds.
  • Some examples with large pushout lattices require further condition checks also not presented.
Searching

You can search by typing in specific second and third betti numbers and the greatest integer divisor of the spin class. These invariants are the most significant in topological classification results. Some more involved searching can be done from the address bar. For example:

. 1 .