Taking Z^4(delta)_Z^4 to represent a 4 dimensional plumbing body, that exists in a 1-handle torus structure with attached circles of new 2-handles. Each 2-handle represents a free factor association matrix aligned with a given 3-manifold over Z^3. Each 1-handle attachment is denoted over Z by an algebraically canceled, free factor cobordism and can be proved using simple homotopy geometric analysis.

If the homotopy of the plumbing manifold classification fails, then a clear recovery would be substituting the band-summation of the 1-handle attachment with a meridian along a suitable band. Such a method would increase uncertainty in the overall plumbing calculations but this can be negated by using a h-cobordism of 4-manifolds, utilizing the Curtis variation for Z.

If the plumbing body is in the form of an immersed Whitney disk, each pair is disjointed by its associated sphere unions. A Kirby diagram of the plumbing manifold structure shows a sphere disjoint that can be denoted as B(j)>K^76_P(Z) across delta. A fundamental group pi(1) of the plumbing structure gives a self-intersection common to Whitney disks that is relative to the handle adjunct of the original torus.

If the Z property of the plumbing body is contractible, then each handle cancels algebraically and gives a defined homology. Such a homology is represented by spheres denoted A(i) and B(i) and will produce further intersection points along the fixed immersed Whitney disk.

A theorem that follows on from homeomorphic plumbing pair transformations is that of Akbulut corks and exotic R^4s. Each of these transformations represents an open subset that has a range of associated linear properties that are explicit in the torus body.

If there exists a smooth cobordism structure with an involution of delta Y across the plumbing body, the calculations can be simplified to give a definite manifold R across the exotic R^4 frame. Each simplification represents a visible Akbulut cork and a subset arrangement commonly found with nontrivial R involution.

A sufficiently isotopic diffeomorphism over the complex surface of the plumbing body has a series of interchanging CP^2-summands that are framed by their tracing structure. The contractible Akbulut cork of the Mazur plumbing manifold will in this case allow no control over complexity handles and realize its role as a linked delta-Y body form. The effect of this on the smooth cobordism structure will be to define the default manifold body as a nontrivial example of definite plumbing body forms.

Source by James McVeigh

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