2-adic Orlik Solomon Algebra Of A Class Of Graph Arrangement And Two-dimensional Non-centralarrangements

The combination and topological invariants of hyperplane arrangement are one of the important topics in the study of hyperplane arrangement.In 2001,M.Falk[2]gave the concept and partial research results of k-adic Orlik-Solomon algebra.Because of k=1,it is the Orlik-Solomon algebra and pointed out that it is also a series of important combination and topological invariants of the hyperplane arrangement.And M.Falk raised an open question about this invariant:PROBLEM 4.1.Calculate the dimension of Akp in terms of the underlying matroid G.M.Falk pointed out:k=2,it has not been resolved.In this paper,we study the 2-adic Orlik Solomon algebra of hyperplane arrangments,and obtain dimension formula of the fourth term and the first four terms of the 2-adic Orlik Solomon algebra of hyperplane arrangement associated with an n-rank wheel graph:dim OS21=2t(n t),t=0,1,2,3,4.And find that n-rank wheel graph is not quadratic.At the same time,we obtain the dimension formula of the fourth term of the 2-adic Orlik Solomon algebra of two classes of linear arrangements.In case of the matrices of the n-rank wheel graph and the two classes of linear arrangements,the results partially answer an open question raised by M.Falk.Meanwhile,we have a guess about the dim-ension of the 2-adic Orlik Solomon algebra of the n-rank wheel graph:dim OS2t=2t(n t),t=0,1,2,…,n;dim OS2t=0,t=n+1,n+2,…,2n?...