Abstract
In this first chapter, we introduce the reader to the main characters of our story, namely, the Morse functions, and we describe the properties which make them so useful. We describe their very special local structure (Morse lemma) and then we show that there are plenty of them around.
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- 1.
This happens because the condition V t (0) = 0 ∀t implies that there exists r > 0 with the property that Ψ t (x) ∈ W, ∀ | x | < r, and ∀ t ∈ [0, 1]. Loosely speaking, if a point x is not very far from the stationary point 0 of the flow Ψ t , then in one second it cannot travel very far along this flow.
- 2.
The reader familiar with the basics of commutative algebra will most certainly recognize that this step of the proof is in fact Nakayama’s lemma in disguise.
- 3.
Thom refers to our nonresonant Morse functions as excellent.
- 4.
See Exercise 6.22 and its solution on page 304.
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© 2012 Springer Science+Business Media, LLC
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Nicolaescu, L. (2012). Morse Functions. In: An Invitation to Morse Theory. Universitext. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-1105-5_1
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DOI: https://doi.org/10.1007/978-1-4614-1105-5_1
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