Index 1

Partitions

Files containing the partitions,
for manifolds of index 1, that annihilate the possible determinants corresponding to the multigraphs.

Mathematica Notebook

Mathematica Notebook that determines the possible isotropy weights

Results

In the index-1 case one can conclude that there are no possible circle actions.

Partitions

Running the c++ files for index 1, we obtain the lists of partitions that annihilate the determinants associated to the multigraphs. These partitions are stored in the files OutputPart#.txt, where # refers to the number of the multigraph. The compressed files can be obtained below.

Partitions

Mathematica Notebook

This Mathematica notebook determines the possible isotropy weights for circle actions on a 8-dimensional manifold of index 1 with 6 fixed points that extends to a 2-torus action. It is divided in several parts described below.

Mathematica Notebook

Part 3

Imports the partition files OutputPart#.txt and sorts the partitions according to the rank of the matrices, dividing them into two sets: those that originate singular matrices with maximal rank, and those that originate matrices of lower rank. In the first case, Part 3 also selects the partitions that originate matrices with null spaces which contain vectors with positive components. It saves the first type of partitions in the files Part#toprk.txt and the others in the files Part#smallrk.txt.

Part 4

Considers the first type of partitions producing a list of the corresponding isotropy weights, checking if they satisfy the polynomial equations in (1.1) of [GS].

Part 5

Considers the second type of partitions:
a) Selects those that originate matrices whose null spaces contain vectors with positive entries.
b) Produces the list of the corresponding isotropy weights that satisfy the polynomial equations in (1.1) of [GS].
c) Selects those isotropy weights that satisfy several necessary properties such as: at each fixed point the isotropy weights must be coprime integers; at a fixed point of index 2i the first i weights are negative and the others are positive; if there are 3 multiple edges between two vertices then the corresponding weights must be coprime ([GS, Lemma 8.2.(1)].

At each step, the resulting lists of isotropy weights are saved in different files so that it is easy to verify which isotropy weights are discarded with each test performed.

Part 6

Concludes that there are no possible isotropy weights in the index-1 case.

Results

In the index-1 case none of the potential isotropy weights satisfies all the necessary conditions and so one concludes that there are no possible circle actions with the prescribed requirements.