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edited May 24 2012 at 18:37 |
I think that Daniele and Sergey have, between the two pretty much answered my question. However, I would like to add the following:
Regarding TOP Morse theory, in [Mor1959] M. Morse laid the foundations of the theory of topological non-degenerate functions, and proved the TOP Morse inequalities.
Also, in [Kui1961] the topological version of the Reeb-Milnor theorem for DIFF manifolds is proven. That is, a TOP n-dimensional closed manifold admiting a TOP Morse function having exactly two non-degenerate critical points is homeomorphic to the n-sphere.
Another source containing further results for topological manifolds via TOP Morse theory that parallel those obtained for differentiable manifolds is J. Cantwell's paper [Can1967].
Kirby and Siebenmann's "Foundational essays on topological manifolds, smoothings, and triangulations is freely available here. In section 3 of Essay III (p. 80) they define Morse functions in the DIFF and TOP categories for manifolds possibly with boundary.
Finally, a simple example examples of a non-differentiable TOP Morse function is are easily found. For example, the absolute value function on $\mathbb{R}$, $x\rightarrow |x|$. The origin is a non-degenerate critical point. Also the height function restricted to the double cone (i.e., the space formed by the cones $x^{2}+y^{2}=(z\pm1)^{2}$) has exactly two non-degenerate critical points (the tips of the cones). Regarding PL Morse theory, J. Harer's slides contain an interesting approach using homology. In particular, a PL Morse function is defined using Betti numbers.
REFERENCES:
[Cant1967] J. Cantwel, Topological non-degenerate functions, Tohoku Math. Journ., 20 (1968), 120-125.
[Kui1961] N.H. Kuiper, A continuous function with two critical points, Bull. Amer. Math. Soc., 67(1961), 281-285.
[Mor1959] M. Morse, Topological non-degenerate functions on a compact manifold $M$, Journal d'Analyse Math., 7 (1959), 189-208.
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edited May 23 2012 at 9:52 |
I think that Daniele and Sergey have, between the two pretty much answered my question. However, I would like to add the following:
Regarding TOP Morse theory, in [Mor1959] M. Morse laid the foundations of the theory of topological non-degenerate functions, and proved the TOP Morse inequalities.
Also, in [Kui1961] the topological version of the Reeb-Milnor theorem for DIFF manifolds is proven. That is, a TOP n-dimensional closed manifold admiting a TOP Morse function having exactly two non-degenerate critical points is homeomorphic to the n-sphere.
Another source containing further results for topological manifolds via TOP Morse theory that parallel those obtained for differentiable manifolds is J. Cantwell's paper [Can1967].
Kirby and Siebenmann's "Foundational essays on topological manifolds, smoothings, and triangulations is freely available here. In section 3 of Essay III (p. 80) they define Morse functions in the DIFF and TOP categories for manifolds possibly with boundary.
Finally, a simple example of a non-differentiable TOP Morse function is the absolute value function on $\mathbb{R}$, $x\rightarrow |x|$. The origin is a non-degenerate critical point. Regarding PL Morse theory, J. Harer's slides contain an interesting approach using homology. In particular, a PL Morse function is defined using Betti numbers.
REFERENCES:
[Cant1967] J. Cantwel, Topological non-degenerate functions, Tohoku Math. Journ., 20 (1968), 120-125.
[Kui1961] N.H. Kuiper, A continuous function with two critical points, Bull. Amer. Math. Soc., 67(1961), 281-285.
[Mor1959] M. Morse, Topological non-degenerate functions on a compact manifold $M$, Journal d'Analyse Math., 7 (1959), 189-208.
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edited May 23 2012 at 8:30 |
I think that Daniele and Sergey have, between the two pretty much answered my question. However, I would like to add the following:
Regarding TOP Morse theory, in [Mor1959] M. Morse laid the foundations of the theory of topological non-degenerate functions, and proved the TOP Morse inequalities.
Also, in [Kui1961] the topological version of the Reeb-Milnor theorem for DIFF manifolds is proven. That is, a TOP n-dimensional manifold admiting a TOP Morse function having exactly two non-degenerate critical points is homeomorphic to the n-sphere.
Another source containing further results for topological manifolds via TOP Morse theory that parallel those obtained for differentiable manifolds is J. Cantwell's paper [Can1967].
Finally,
Kirby and Siebenmann's "Foundational essays on topological manifolds, smoothings, and triangulations is freely available here. In section 3 of Essay III (p. 80) they define Morse functions in the DIFF and TOP categories for manifolds possibly with boundary.
Finally, a simple example of a non-differentiable TOP Morse function is the absolute value function on $\mathbb{R}$, $x\rightarrow |x|$. The origin is a non-degenerate critical point. Regarding PL Morse theory, J. Harer's slides contain an interesting approach using homology. In particular, a PL Morse function is defined using Betti numbers.
REFERENCES:
[Cant1967] J. Cantwel, Topological non-degenerate functions, Tohoku Math. Journ., 20 (1968), 120-125.
[Kui1961] N.H. Kuiper, A continuous function with two critical points, Bull. Amer. Math. Soc., 67(1961), 281-285.
[Mor1959] M. Morse, Topological non-degenerate functions on a compact manifold $M$, Journal d'Analyse Math., 7 (1959), 189-208.
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edited May 23 2012 at 8:18 |
I think that Daniele and Sergey have, between the two pretty much answered my question. However, I would like to add the following:
Regarding TOP Morse theory, in [Mor1959] M. Morse laid the foundations of the theory of topological non-degenerate functions, and proved the TOP Morse inequalities.
Also, in [Kui1961] the topological version of the Reeb-Milnor theorem for DIFF manifolds is proven. That is, a TOP n-dimensional manifold admiting a TOP Morse function having exactly two non-degenerate critical points (degenerate or not) is homeomorphic to the n-sphere.
Another source containing further results for topological manifolds via TOP Morse theory that parallel those obtained for differentiable manifolds is J. Cantwell's paper [Can1967].
Finally, Kirby and Siebenmann's "Foundational essays on topological manifolds, smoothings, and triangulations is freely available here. In section 3 of Essay III (p. 80) they define Morse functions in the DIFF and TOP categories. Regarding PL Morse theory, J. Harer's slides contain an interesting approach using homology. In particular, a PL Morse function is defined using Betti numbers.
REFERENCES:
[Cant1967] J. Cantwel, Topological non-degenerate functions, Tohoku Math. Journ., 20 (1968), 120-125.
[Kui1961] N.H. Kuiper, A continuous function with two critical points, Bull. Amer. Math. Soc., 67(1961), 281-285.
[Mor1959] M. Morse, Topological non-degenerate functions on a compact manifold $M$, Journal d'Analyse Math., 7 (1959), 189-208.
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edited May 23 2012 at 7:41 |
I think that Daniele and Sergey have, between the two pretty much answered my question. However, I would like to add the following:
Regarding TOP Morse theory, in [Mor1959] M. Morse laid the foundations of the theory of topological non-degenerate functions, and proved the TOP Morse inequalities.
Also, in [Kui1961] the topological version of the Reeb-Milnor theorem for DIFF manifolds is proven. That is, a TOP n-dimensional manifold admiting a TOP Morse function having exactly two critical points (degenerate or not) is homeomorphic to the n-sphere.
Another source containing further results for topological manifolds via TOP Morse theory that parallel those obtained for differentiable manifolds is J. Cantwell's paper [Can1967].
Finally, Kirby and Siebenmann's "Foundational essays on topological manifolds, smoothings, and triangulations is freely available here. Section In section 3 of Essay III (p. 80). They 80) they define more Morse functions in the DIFF and TOP categories. Regarding PL Morse theory, J. Harer's slides containg contain an interesting approach using homology. In particular, a PL Morse function is defined using Betti numbers.
REFERENCES:
[Cant1967] J. Cantwel, Topological non-degenerate functions, Tohoku Math. Journ., 20 (1968), 120-125.
[Kui1961] N.H. Kuiper, A continuous function with two critical points, Bull. Amer. Math. Soc., 67(1961), 281-285.
[Mor1959] M. Morse, Topological non-degenerate functions on a compact manifold $M$, Journal d'Analyse Math., 7 (1959), 189-208.
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answered May 23 2012 at 7:08 |
I think that Daniele and Sergey have, between the two pretty much answered my question. However, I would like to add the following:
Regarding TOP Morse theory, in [Mor1959] M. Morse laid the foundations of the theory of topological non-degenerate functions, and proved the TOP Morse inequalities.
Also, in [Kui1961] the topological version of the Reeb-Milnor theorem for DIFF manifolds is proven. That is, a TOP n-dimensional manifold admiting a TOP Morse function having exactly two critical points (degenerate or not) is homeomorphic to the n-sphere.
Another source containing further results for topological manifolds via TOP Morse theory that parallel those obtained for differentiable manifolds is J. Cantwell's paper [Can1967].
Finally, Kirby and Siebenmann's "Foundational essays on topological manifolds, smoothings, and triangulations is freely available here. Section 3 of Essay III (p. 80). They define more functions in the DIFF and TOP categories. Regarding PL Morse theory, J. Harer's slides containg an interesting approach using homology. In particular, a PL Morse function is defined using Betti numbers.
REFERENCES:
[Cant1967] J. Cantwel, Topological non-degenerate functions, Tohoku Math. Journ., 20 (1968), 120-125.
[Kui1961] N.H. Kuiper, A continuous function with two critical points, Bull. Amer. Math. Soc., 67(1961), 281-285.
[Mor1959] M. Morse, Topological non-degenerate functions on a compact manifold $M$, Journal d'Analyse Math., 7 (1959), 189-208.
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