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Under certain conditions, (Morse-Smale being one, but not sufficient) the stratification by unstable manifolds of a Morse flow on a compact manifolds gives a cellular decomposition; see the paper On the Space of Trajectories of a Generic Vector Field by Burghelea & co, and the paper On Moduli Spaces and CW Structures Arising from Morse Theory On Hilbert Manifolds of Lizhen Qin.

Having a cellular decomposition, does not make reading the boundary map any easier. However, the boundary map has a dynamical description, and the complex with this description is usually referred to as the Floer complex of a Morse-Smale gradient flow.

If you are interested in $\mathbb{Z}$-coefficients things can be tricky due to various sign issues. Over $\mathbb{Z}/2$ these sign issues are no longer relevant and the computations are much easier.

For (real) Grassmanians, it is relatively easy to compute their $\mathbb{Z}/2$-Betti numbers using Morse theory. A particularly nice paper is Integrable Gradient Flows and Morse Theory by Dynnikov and Veselov.

To see how a combination of Morse theory and the technques of Harvey-Lawson work, check my paper Schubert calculus on the Grassmannian of hermitian lagrangian spaces (doi:10.1016/j.aim.2010.02.003) where I use Morse theoretic techniques to produce a (real) Schubert calculus on $U(n)$. In particular, this yields a geometric description of the integral cohomology ring of $U(n)$ Arguments similar to the ones in this paper can be used to compute the integral homology of any grassmanian, real or complex, by Morse theoretic means. These include the projective space $\mathbb{RP}^n$.

For more on this subject I would suggest you have a look at Sec. 2.5 and Chaper 4 of the 2nd Edition of my book, An Invitation to Morse Theory where I go through these issues in great detail.

Under certain conditions, (Morse-Smale being one, but not sufficient) the stratification by unstable manifolds of a Morse flow on a compact manifolds gives a cellular decomposition; see the paper On the Space of Trajectories of a Generic Vector Field by Burghelea & co, and the paper On Moduli Spaces and CW Structures Arising from Morse Theory On Hilbert Manifolds of Lizhen Qin.

Having a cellular decomposition, does not make reading the boundary map any easier. However, the boundary map has a dynamical description, and the complex with this description is usually referred to as the Floer complex of a Morse-Smale gradient flow.

If you are interested in $\mathbb{Z}$-coefficients things can be tricky due to various sign issues. Over $\mathbb{Z}/2$ these sign issues are no longer relevant and the computations are much easier.

For (real) Grassmanians, it is relatively easy to compute their $\mathbb{Z}/2$-Betti numbers using Morse theory. A particularly nice paper is Integrable Gradient Flows and Morse Theory by Dynnikov and Veselov.

To see how a combination of Morse theory and the technques of Harvey-Lawson work, check my paper Schubert calculus on the Grassmannian of hermitian lagrangian spaces (doi:10.1016/j.aim.2010.02.003) where I use Morse theoretic techniques to produce a (real) Schubert calculus on $U(n)$. In particular, this yields a geometric description of the integral cohomology ring of $U(n)$ Arguments similar to the ones in this paper can be used to compute the integral homology of any grassmanian, real or complex, by Morse theoretic means. These include the projective space $\mathbb{RP}^n$.

For more on this subject I would suggest you have a look at Sec. 2.5 and Chaper 4 of the 2nd Edition of my book, An Invitation to Morse Theory where I go through these issues in great detail.

Under certain conditions, (Morse-Smale being one, but not sufficient) the stratification by unstable manifolds of a Morse flow on a compact manifolds gives a cellular decomposition; see this paper by Burghelea & co and this paper of Lizhen Qin.

Having a cellular decomposition, does not make reading the boundary map any easier. However, the boundary map has a dynamical description, and the complex with this description is usually referred to as the Floer complex of a Morse-Smale gradient flow.

If you are interested in $\mathbb{Z}$-coefficients things can be tricky due to various sign issues. Over $\mathbb{Z}/2$ these sign issues are no longer relevant and the computations are much easier.

For  (real) Grassmanians, it is relatively easy to compute their $\mathbb{Z}/2$-Betti numbers using Morse theory. A particularly nice paper is this one by Dynnikov & Vesselov.

To see how a combination of Morse theory and the technques of Harvey-Lawson work, check this paper where I use Morse theoretic techniques to produce a (real) Schubert calculus on $U(n)$. In particular, this yields a geometric description of the integral cohomology ring of $U(n)$ Arguments similar to the ones in this paper can be used to compute the integral homology of any grassmanian, real or complex, by Morse theoretic means. These include the projective space $\mathbb{RP}^n$.

For more on this subject I would suggest you have a look at Sec. 2.5 and Chaper 4 of the 2nd Edition of my book, An Invitation to Morse Theory where I go through these issues in great detail.

Under certain conditions, (Morse-Smale being one, but not sufficient) the stratification by unstable manifolds of a Morse flow on a compact manifolds gives a cellular decomposition; see the paper On the Space of Trajectories of a Generic Vector Field by Burghelea & co, and the paper On Moduli Spaces and CW Structures Arising from Morse Theory On Hilbert Manifolds of Lizhen Qin.

Having a cellular decomposition, does not make reading the boundary map any easier. However, the boundary map has a dynamical description, and the complex with this description is usually referred to as the Floer complex of a Morse-Smale gradient flow.

If you are interested in $\mathbb{Z}$-coefficients things can be tricky due to various sign issues. Over $\mathbb{Z}/2$ these sign issues are no longer relevant and the computations are much easier.

For (real) Grassmanians, it is relatively easy to compute their $\mathbb{Z}/2$-Betti numbers using Morse theory. A particularly nice paper is Integrable Gradient Flows and Morse Theory by Dynnikov and Veselov.

To see how a combination of Morse theory and the technques of Harvey-Lawson work, check my paper Schubert calculus on the Grassmannian of hermitian lagrangian spaces (doi:10.1016/j.aim.2010.02.003) where I use Morse theoretic techniques to produce a (real) Schubert calculus on $U(n)$. In particular, this yields a geometric description of the integral cohomology ring of $U(n)$ Arguments similar to the ones in this paper can be used to compute the integral homology of any grassmanian, real or complex, by Morse theoretic means. These include the projective space $\mathbb{RP}^n$.

For more on this subject I would suggest you have a look at Sec. 2.5 and Chaper 4 of the 2nd Edition of my book, An Invitation to Morse Theory where I go through these issues in great detail.

Under certain conditions, (Morse-Smale being one, but not sufficient) the stratification by unstable manifolds of a Morse flow on a compact manifolds gives a cellular decomposition; see this paper by Burghelea & co and this paper of Lizhen Qin.

Having a cellular decomposition, does not make reading the boundary map any easier. However, the boundary map has a dynamical description, and the complex with this description is usually referred to as the Floer complex of a Morse-Smale gradient flow.

If you are interested in $\mathbb{Z}$-coefficients things can be tricky due to various sign issues. Over $\mathbb{Z}/2$ these sign issues are no longer relevant and the computations are much easier.

For (real) Grassmanians, it is relatively easy to compute their $\mathbb{Z}/2$-Betti numbers using Morse theory. A particularly nice paper is this one by Dynnikov & Vesselov.

To see how a combination of Morse theory and the technques of Harvey-Lawson work, check this paper where I use Morse theoretic techniques to produce a (real) Schubert calculus on $U(n)$. In particular, this yields a geometric description of the integral cohomology ring of $U(n)$ Arguments similar to the ones oin this apper can be used to compute the integral homology of any grassmanian real or complex by Morse theoretic means. This include the projective space $\mathbb{RP}^n$.

For more on this subject I would suggest you have a look at Sec. 2.5 and Chaper 4 of the 2nd Edition of my book, An Invitation to Morse Theory where I go through these issues in great detail.

Under certain conditions, (Morse-Smale being one, but not sufficient) the stratification by unstable manifolds of a Morse flow on a compact manifolds gives a cellular decomposition; see this paper by Burghelea & co and this paper of Lizhen Qin.

Having a cellular decomposition, does not make reading the boundary map any easier. However, the boundary map has a dynamical description, and the complex with this description is usually referred to as the Floer complex of a Morse-Smale gradient flow.

If you are interested in $\mathbb{Z}$-coefficients things can be tricky due to various sign issues. Over $\mathbb{Z}/2$ these sign issues are no longer relevant and the computations are much easier.

For (real) Grassmanians, it is relatively easy to compute their $\mathbb{Z}/2$-Betti numbers using Morse theory. A particularly nice paper is this one by Dynnikov & Vesselov.

To see how a combination of Morse theory and the technques of Harvey-Lawson work, check this paper where I use Morse theoretic techniques to produce a (real) Schubert calculus on $U(n)$. In particular, this yields a geometric description of the integral cohomology ring of $U(n)$ Arguments similar to the ones in this paper can be used to compute the integral homology of any grassmanian, real or complex, by Morse theoretic means. These include the projective space $\mathbb{RP}^n$.

For more on this subject I would suggest you have a look at Sec. 2.5 and Chaper 4 of the 2nd Edition of my book, An Invitation to Morse Theory where I go through these issues in great detail.