Is the Transversal Intersection of Manifolds a Manifold?

  1. WWGD

    WWGD 1,323
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    Hi, All:

    Given manifolds M,N (both embedded in $R^n$, intersecting each other transversally,

    so that their intersection has dimension >=1 ( i.e. n -(Dim(M)-Dim(N)>1) is the intersection

    a manifold?

    Thanks.
     
  2. jcsd
  3. quasar987

    quasar987 4,770
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    Yes, according to the "canonical form theorem" for a transverse intersection, if X,Y are submanifolds of the n-manifold M that intersect transversally "of dimension k", and if p is a point of intersection, there is a coordinate nbhd of p in M such that X n Y corresponds to R^k in R^n under the coordinate map.
     
  4. WWGD

    WWGD 1,323
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    Thanks, Quasar; any chance you have a ref?
     
  5. quasar987

    quasar987 4,770
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    See the book Differential manifolds by Antoni Kosinski where the stronger result is proved that actually, there is a coordinate chart around p in which X corresponds to R^r x {0} while Y corresponds to {0} x R^s, so what X n Y, of course, corresponds to {0} x R^k x {0} (where r=dim(X), s=dim(Y), and k=(r+s)-).

    But surely the theorem can also be found in Differential Topology by Guillemin & Pollack and possibly in the book of the same name by M. Hirsh
     
    Last edited: Jul 4, 2011
  6. WWGD:

    Not to quibble too much, but I have seen two main definitions of transversality used,

    and I wondered which one you are using:

    1) First and weaker of the two, states that if submanifolds S,S' of ambient M intersect

    transversely at p, then the (vector space) sum of the tangent spaces TpS and TpS'

    equals TpM , i.e., the sum spans the tangent space of the ambient manifold.

    2) The second and stronger one (stronger in that it excludes some cases of 1) , is

    that each point p of intersection has a neighborhood Up with Phi(Up)=

    (x1,x2,..,xn,0,0,..,0) (Phi is a chart map). This excludes, e.g., Sin(1/x) and the

    x-axis.

    BTW, the Mathworld entry "http://mathworld.wolfram.com/HomologyIntersection.html" seems to agree in a weaker

    sense, in that it states that cycles (homology classes) that intersect transversely are also cycles ,and it gives

    the example of y=x^2 and the x-axis as an example of a non-transverse intersection in the sense 2) above,

    showing how the intersection is unstable, in that a small perturbation --e.g., moving y=x^2 upwards changes

    the (algebraic) intersection number .
     
    Last edited: Jul 14, 2011
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