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(1). Some experts tell me that Laudenbach's paper is incomplete and contains gaps.

(2). The result you seek can be deduced in the following papers by Lizhen Qin (disclaimer: he was my student):

On moduli spaces and CW structures arising from Morse theory on Hilbert manifolds. J. Topol. Anal. 2 (2010), no. 4, 469–526

An application of topological equivalence to Morse theory. arXiv:1102.2838

In fact, what you ask can be deduced from the first of these papers which handles a metric that is flat near the critical points. The second paper shows that one can actually use any metric such that the function is Morse-Smale.

Alternatively, there is a paper of Burghelea and Haller which also handles the case of a metric that is flat near the critical points. Lizhen tells me that the Burghelea-Haller paper is correct and one can deduce the desired result from their methods.

(The reason I am ranting about this is that I believe this area to be a hotbed of papers containing gaps--with no disrespect to the authors of those papers intended.)

(1). Some experts tell me that Laudenbach's paper is incomplete and contains gaps.

I will retract this for now. I do recall being told this, but I am not aware at this point in time where the gaps in his paper are, if any. (I do stand by my belief that a number papers in this area are incomplete.)

(2). The result you seek can be deduced in the following papers by Lizhen Qin (disclaimer: he was my student):

On moduli spaces and CW structures arising from Morse theory on Hilbert manifolds. J. Topol. Anal. 2 (2010), no. 4, 469–526

An application of topological equivalence to Morse theory. arXiv:1102.2838

In fact, what you ask can be deduced from the first of these papers which handles a metric that is flat near the critical points. The second paper shows that one can actually use any metric such that the function is Morse-Smale.

Alternatively, there is a paper of Burghelea and Haller which also handles the case of a metric that is flat near the critical points. Lizhen tells me that the Burghelea-Haller paper is correct and one can deduce the desired result from their methods.

(The reason I am ranting about this is that I believe this area to be a hotbed of papers containing gaps--with no disrespect to the authors of those papers intended.)

(1). Some experts tell me that Laudenbach's paper is incomplete and contains gaps.

(2). The result you seek can be deduced in the following papers by Lizhen Qin (disclaimer: he was my student):

On moduli spaces and CW structures arising from Morse theory on Hilbert manifolds. J. Topol. Anal. 2 (2010), no. 4, 469–526

An application of topological equivalence to Morse theory. arXiv:1102.2838

In fact, what you ask can be deduced from the first of these papers which handles a metric that is flat near the critical points. The second paper shows that one can actually use any metric such that the function is Morse-Smale.

Alternatively, there is a paper of Burghelea and Haller which also handles the case of a metric that is flat near the critical points. (Lizhen tells me that the Burghelea-Haller paper is correct.)

(The reason I am ranting about this is that I believe this area to be a hotbed of papers containing gaps--with no disrespect to the authors of those papers intended.)

(1). Some experts tell me that Laudenbach's paper is incomplete and contains gaps.

(2). The result you seek can be deduced in the following papers by Lizhen Qin (disclaimer: he was my student):

On moduli spaces and CW structures arising from Morse theory on Hilbert manifolds. J. Topol. Anal. 2 (2010), no. 4, 469–526

An application of topological equivalence to Morse theory. arXiv:1102.2838

In fact, what you ask can be deduced from the first of these papers which handles a metric that is flat near the critical points. The second paper shows that one can actually use any metric such that the function is Morse-Smale.

Alternatively, there is a paper of Burghelea and Haller which also handles the case of a metric that is flat near the critical points. Lizhen tells me that the Burghelea-Haller paper is correct and one can deduce the desired result from their methods.

(The reason I am ranting about this is that I believe this area to be a hotbed of papers containing gaps--with no disrespect to the authors of those papers intended.)

(1). Some experts tell me that Laudenbach's paper is incomplete and contains gaps.

(2). The result you seek can be deduced in the following papers by Lizhen Qin (disclaimer: he was my student):

On moduli spaces and CW structures arising from Morse theory on Hilbert manifolds. J. Topol. Anal. 2 (2010), no. 4, 469–526

An application of topological equivalence to Morse theory. arXiv:1102.2838

In fact, what you ask can be deduced from the first of these papers which handles a metric that is flat near the critical points. The second paper shows that one can actually use any metric such that the function is Morse-Smale.

Alternatively, there is a paper of Burghelea which also handles the case of a metric that is flat near the critical points. (Lizhen tells me that Burghelea's paper is correct.)

(The reason I am ranting about this is that I believe this area to be a hotbed of papers containing gaps--with no disrespect to the authors of those papers intended.)

(1). Some experts tell me that Laudenbach's paper is incomplete and contains gaps.

(2). The result you seek can be deduced in the following papers by Lizhen Qin (disclaimer: he was my student):

On moduli spaces and CW structures arising from Morse theory on Hilbert manifolds. J. Topol. Anal. 2 (2010), no. 4, 469–526

An application of topological equivalence to Morse theory. arXiv:1102.2838

In fact, what you ask can be deduced from the first of these papers which handles a metric that is flat near the critical points. The second paper shows that one can actually use any metric such that the function is Morse-Smale.

Alternatively, there is a paper of Burghelea and Haller which also handles the case of a metric that is flat near the critical points. (Lizhen tells me that the Burghelea-Haller paper is correct.)

(The reason I am ranting about this is that I believe this area to be a hotbed of papers containing gaps--with no disrespect to the authors of those papers intended.)