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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Summer 2016 • Question 4

Question

Math 6510 • Differentiable Manifolds Summer 2016 • Question 4

Let \(M,N,P\) be smooth manifolds without boundary and \(F:M\times P\to N\) a smooth map which is transverse to a submanifold \(Q\subset N\). Prove the Transversality Theorem: for almost every \(p\in P\) the map \(F_p:M\to N\) defined by \(F_p(x)=F(x,p)\) is transverse to \(Q\). If you’d like, you can consider the special case when \(M,Q\) are submanifolds of \(P=N=\mathbb {R}^n\) and \(F(x,p)=x+p\).

Raw LaTeX

Let \(M,N,P\) be smooth manifolds without boundary and
\(F:M\times P\to N\) a smooth map which is transverse to a submanifold
\(Q\subset N\). Prove the Transversality Theorem: for almost every \(p\in P\) the
map \(F_p:M\to N\) defined by \(F_p(x)=F(x,p)\) is transverse to \(Q\). If you'd like,
you can consider the special case when \(M,Q\) are submanifolds of
\(P=N=\R^n\) and \(F(x,p)=x+p\).

Topic Tags

Homotopy and Degree on Manifolds Sard's Theorem and Generic Transversality Smooth Manifolds and Embedded Submanifolds Transversality and Intersection Theory