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Hi, everyone :
I am reading about "Nodal singularities" ,and I have not been able to find
a clear def. of what they are. Neither of the standard sources: Wiki, Google,
or some of the books I have checked, has a clear explanation. Here is the
context:
We are trying to show that we can represent elements of H^2(M;Z), in a 4-D,
orientable manifold M, by embedded submanifolds.
An example mentioned is that of the coordinate planes Z1=0 and Z2=0 , meeing
at the origin. I guess these are the coordinate axes(planes) in C^2 ~ R^4 (C=complexes)
Now, mention is made that " a double-point is isomorphic to the simple nodal
singularity :
Z1*Z2=0
meeting at the origin. Now, I have an idea of what is going on, but I would like
to have actual definitions of "double-points" (equiv: n-point ) singularity, and
of nodal singularity. I believe that the singularity has to see with having a corner,
a sharp corner that is not smooth, tho I am not clear on what the issue is with
nodal singularity.
Also, it is mentioned that the singularities can be perturbed away slightly, at the
price of increasing the genus. Is this the standard def. of genus, i.e., the genus
of a surface is twice the number of non-separating loops on the surface ( or, equiv.
, I guess, non-separating manifolds?)
Thanks For Any Help.
I am reading about "Nodal singularities" ,and I have not been able to find
a clear def. of what they are. Neither of the standard sources: Wiki, Google,
or some of the books I have checked, has a clear explanation. Here is the
context:
We are trying to show that we can represent elements of H^2(M;Z), in a 4-D,
orientable manifold M, by embedded submanifolds.
An example mentioned is that of the coordinate planes Z1=0 and Z2=0 , meeing
at the origin. I guess these are the coordinate axes(planes) in C^2 ~ R^4 (C=complexes)
Now, mention is made that " a double-point is isomorphic to the simple nodal
singularity :
Z1*Z2=0
meeting at the origin. Now, I have an idea of what is going on, but I would like
to have actual definitions of "double-points" (equiv: n-point ) singularity, and
of nodal singularity. I believe that the singularity has to see with having a corner,
a sharp corner that is not smooth, tho I am not clear on what the issue is with
nodal singularity.
Also, it is mentioned that the singularities can be perturbed away slightly, at the
price of increasing the genus. Is this the standard def. of genus, i.e., the genus
of a surface is twice the number of non-separating loops on the surface ( or, equiv.
, I guess, non-separating manifolds?)
Thanks For Any Help.
