I'm confused about your 2nd question. Milnor's Morse theory does use a Riemann metric -- he's using the gradient flow. To define the gradient he needs an inner product on the tangent spaces. Without the gradient flow you don't have the cellular attaching maps.

Regarding your 1st question, there's something that's much better than a CW-structure. A Morse function gives a handle decomposition of the manifold. This can be used to talk about the smooth structure. A CW-decomposition is relatively degenerate in comparison. The handle decomposition is described in Milnor's h-cobordism notes.

Taking your 1st question more seriously, you run into technical problems. The gradient flows do not give you a CW-decomposition of the manifold -- for example consider Milnor's Morse Theory example of a torus with height function. The Morse function and its gradient flows gives you a genuine 1-skeleton (figure-8). But the attaching map for the 2-cell (to the figure-8) is not a continuous function if you use the gradient flow -- all points except for two go to the global minimum for the height function. This shows you the kind of problems you encounter if you want to produce a genuine CW-decomposition of the manifold.

So if you're not going to use solely the gradient flows to define the attaching maps for the proposed CW-decomposition, what do you allow? All smooth manifolds admit CW-decompositions so if you allow sufficient tweaking you can of course fix this construction but if you allow "too much" tweaking, the CW-decomposition won't be an invariant of the Morse function.

edit: Here is a way to tweak the process. The gradient flow does give you a genuine 1-skeleton. So take a regular neighbourhood of the 1-skeleton, and perturb the original vector field in this regular neighbourhood to point in towards the 1-skeleton. This makes the 2-cell attaching maps continuous (terminating in a finite amount of time). Then take a regular neighbourhood of the 2-skeleton, and perturb the vector field to point in towards the 2-skeleton. Again, you get flow lines terminating in finite-time so you get genuine 2-cell attaching maps. The problem with this is you're getting a CW-decomposition but it depends on more than the Morse function as you need to choose smooth regular neighbourhoods of the skeleta.

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I'm confused about your 2nd question. Milnor's Morse theory does use a Riemann metric -- he's using the gradient flow. To define the gradient he needs an inner product on the tangent spaces. Without the gradient flow you don't have the cellular attaching maps.

Regarding your 1st question, there's something that's much better than a CW-structure. A Morse function gives a handle decomposition of the manifold. This can be used to talk about the smooth structure. A CW-decomposition is relatively degenerate in comparison. The handle decomposition is described in Milnor's h-cobordism notes.

Taking your 1st question more seriously, you run into technical problems. The gradient flows do not give you a CW-decomposition of the manifold -- for example consider Milnor's Morse Theory example of a torus with height function. The Morse function and its gradient flows gives you a genuine 1-skeleton (figure-8). But the attaching map for the 2-cell (to the figure-8) is not a continuous function if you use the gradient flow -- all points except for two go to the global minimum for the height function. This shows you the kind of problems you encounter if you want to produce a genuine CW-decomposition of the manifold.

So if you're not going to use solely the gradient flows to define the attaching maps for the proposed CW-decomposition, what do you allow? All smooth manifolds admit CW-decompositions so if you allow sufficient tweaking you can of course fix this construction but if you allow "too much" tweaking, the CW-decomposition won't be an invariant of the Morse function.