Application of Rohklin’s Theorem to Plumbing Manifolds

Application of Rohklin’s Theorem to Plumbing Manifolds




One of the meaningful consequences of applying theorems such as the Rokhlin theorem to invariants for 3-manifolds (in the plumbing sphere), is that high-dimensions have a topological homology on the Z16-invariant. High-dimensions such as four and beyond must be prepared with what is known as ‘additivity of signatures’.

Additivity of signatures involves connecting two 4-manifolds with a non-empty border. This results in a unimodular intersection form, except in the case when the border character is a homology sphere. These similarities are easy to prove as the topological cobordism group Qr(top) orients relative to the isomorphism of the structure.

Since lambda is a homology 3-sphere, the smooth spin, 4-manifold plumbing border is an already intersection form. Using Van der Blij’s lemma corollary we can place the algebraic modulo sign as either 0 or 8. This then allows us to define the Rokhlin invariant as p(1/8)signM (mod 2).

This is a well-defined invariant of lambda and can be placed with a closed spin manifold structure such that the Poincare homology 3-sphere, giving a compatibility coefficient of 1.

Taking N to be equal to a rare spin structure, the oriented cobordism plumbing group has a dimensional disjoint union of modulo lamda and an empty manifold defined by null zero. The Abelian group of this structure has an equivalence class such that the oriented manifold W is oriented according to the expressed border vertices of the plumbing 3-manifold group.

Kirby’s law states that the topology of 4-manifolds has a geometric proof according to its low-dimensional cobordism statements and an isomorphic Thom construction over the plumbing sphere.

consequently using this statement, we can construct a framed submanifold by the differential map of lamda-k modulo. This calculation gives a trivialization of the plumbing graph bundle and an approximation of the 4-manifold sphere such that it resembles the output of the Rokhlin invariant.

Taking the Rohklin invariant as equal to the framed bordant of the plumbing submanifold, the trivialized normal bundles are defined by their border restrictions that provide f:X ->S(m) according to their border weights. The bijection proof of this group structure is equal to the isomorphism of the plumbing graph and has a bordism addition that is obvious when the homotopy identity of lamda is defined.

The global character of the finite polyhedron is an haphazard size defined by the orthogonal group of unimodular quadratic forms. The inertia index has a genus and algebraic geometry that gives the final plumbing monograph required.




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