A proof of the Morse-Bott Lemma

A simple proof of Zariski's Lemma

‎Our aim in this very short note is to show that the proof of the‎ ‎following well-known fundamental lemma of Zariski follows from an‎ ‎argument similar to the proof of the fact that the rational field‎ ‎$mathbb{Q}$ is not a finitely generated $mathbb{Z}$-algebra.

The Morse–Bott–Kirwan condition is local

A Noncommutative Proof of Bott Periodicity

Bott periodicity in K-theory is a rather mysterious object. The classical proofs typically consist of showing that the unitary groups form an Ω-spectrum from which to get a cohomology theory; then showing that that theory is K-theory; and most formidably showing that U(n) is homotopic to U(n+ 2) for all n. However, Cuntz showed that Bott periodicity can be derived in a much simpler way if one r...

The Morse Lemma for Banach Spaces

REMARK. The above theorem is a classical result of Marston Morse in the case that V is finite dimensional and was generalized by the author to the case that F is a Hubert space [ l ] , [3]. The latter proof makes use of operator theory in Hubert space and does not extend in any obvious way to more general Banach spaces. The proof we give below is completely elementary and works for arbitrary V ...

The Morse Lemma on Banach Spaces

Let /: U-*R be a C3 map of an open subset U of a Banach space E. Letp £ U be a critical point of/ (df¡,=0). If £ is a conjugate space (E=F*) we define what it means for/» to be nondegenerate. In this case there is a diffeomorphism y of a neighborhood of p with a neighborhood of 0 E E, y(p)=0 with foy->(x) = id%(x,x)+f(p). In this paper we define a notion of nondegeneracy for critical points of ...