The orientation line of a Morse critical point

Let $(f,X)$ be a Morse--Smale pair on a manifold $M$ and let $p$ be a critical point of index $\lambda (p)$. The orientation line of $p$ is the determinant line $o_p:=\det T_p{W}^u(p)={\mathrm{\Lambda }}^{\lambda (p)}{T}_p{W}^u(p)$ of the tangent space at $p$ of the unstable manifold (Determinant-line orientations of finite-dimensional real vector spaces). An orientation of the critical point is a choice of positive ray in $o_p$; equivalently, since $W^u(p)$ is connected and diffeomorphic to ${\mathbb{R}}^{\lambda (p)}$ (Global stable and unstable manifolds are immersed Euclidean spaces), it is a choice of orientation of the disk $W^u(p)$. No orientation of $M$ and no orientability of $M$ is used; the orientation lines are extra data attached to the critical points.

Here $W^u(p)$ is the unstable set of $p$ under the descending flow (Stable and unstable sets of a critical point), an immersed $\lambda (p)$-dimensional manifold by Global stable and unstable manifolds are immersed Euclidean spaces, with $\lambda (p)$ the Morse index of Nondegenerate critical points, nullity, index, and coindex; the pair $(f,X)$ is Morse--Smale in the sense of Morse--Smale pairs, so $X$ is complete. The unstable manifold is diffeomorphic to ${\mathbb{R}}^{\lambda (p)}$ and hence connected and orientable. Flow invariance (Stable and unstable manifolds are flow invariant) makes $X$ tangent to it: differentiating $t\mapsto {\mathrm{\Phi }}_t(x)\in W^u(p)$ at $t=0$ gives $X_x\in T_x{W}^u(p)$. Because $W^u(p)$ is connected and diffeomorphic to a Euclidean space, a ray in $\det T_p{W}^u(p)$ extends uniquely to a continuous orientation of $W^u(p)$ (pull back to Euclidean space and choose the constant sign matching the ray at $p$); and restricting an orientation of $W^u(p)$ back to $p$ returns the ray: the two descriptions of an orientation of $p$ agree. For $\lambda (p)=0$ the line $o_p$ is ${\mathrm{\Lambda }}^0\{0\}=\mathbb{R}$ and an orientation of $p$ is a choice of one of its two rays, matching the two orientations of a one-point manifold.

The orientation line is attached to $p$, not to $M$: the tangent space $T_p{W}^u(p)$ is defined by the backward-limit set of the flow, and $o_p$ carries no information about an orientation of the ambient manifold. Different critical points may be oriented independently, and replacing the chosen ray by its opposite is the operation of reversing the orientation of $p$ used later.