# Talk:Unimodular lattice

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Field:  Number theory

## Use of the term "lattice" needs uniformization

Although this article defines "lattice" in the manner that it uses the term, the article also refers (as it well should) to the Wikipedia article lattice, which defines lattice in a somewhat different manner, although the two definitions are closely related.

This article, unimodular lattice, defines "lattice" as a free abelian group of finite rank possessing an integral symmetric bilinear form.

The article lattice (group), on the other hand, defines "lattice" as a discrete subgroup of Rn that spans Rn (over the field R). This definition is more general, as it contains no integrality condition. It is also less abstract. (It is also unnecessarily restrictive, since a lattice in Rn need only span a vector subspace of Rn. This restriction does not unnecessarily limit the isomorphism classes of lattices, but it does limit what satisfies the definition of a lattice.)

I believe these two articles ought to be reconciled with each other.Daqu (talk) 22:21, 12 April 2008 (UTC)

• I made a first step by distinguishing "integral lattice" from "lattice" in the definition section. "Unimodular" assumes integral, but "lattice" should not. Eigenbra (talk) 01:41, 17 August 2014 (UTC)

## Use of the term "norm" needs explaining

From the article's statement that the Leech lattice has no vectors of norm 1 or 2, one may infer that the term "norm" is not being used here to mean the length of a vector (since the Leech lattice does have vectors of length 2). Probably "norm" in this article refers to the length squared of a vector. But this is not explained in this article, nor in the Wikipedia entry on norm. I suggest that this be clarified, both here and in the entry for norm.Daqu (talk) 22:00, 12 April 2008 (UTC)

## Applications

I think that the statement "The second cohomology group of a compact simply connected oriented topological 4-manifold is a unimodular lattice" needs to be corrected. This is true only if the manifold is closed, or if its boundary is homology spheres. —Preceding unsigned comment added by 94.194.40.121 (talk) 08:44, 18 May 2011 (UTC)