Existence of fundamental class of $H_i(M,\partial M;R)$ Here $M$ is a compact connected manifold with boundary, how to show there is relative a fundamental class of $H_i(M,\partial M;R)$?
Here fundamental class means an element of $H_n(M,\partial M;R)$ whose image in $H_n(M|x;R)$ is a generator for all $x\in M−\partial M$
 A: The idea is that if the boundary has a collar neighborhood, then the inclusion $(M,\partial M\times[0,\varepsilon))\to (M,\partial M)$ is a homotopy equivalence.  Followed by excision, then $H_n(M,\partial M;R)$ is isomorphic to $H_n(M-\partial M,\partial M\times(0,\varepsilon);R)$.  Now, if $M$ is $R$-orientable, then so is the open submanifold $M-\partial M$, and Lemma 3.27 applies since $(M-\partial M)-(\partial M\times (0,\varepsilon))$ is a compact subset.  The lemma implies that this homology group has a unique generator which is a relative fundamental class.  The fact that it actually is a relative fundamental class of $(M,\partial M)$ is due to naturality.
In particular, let $x\in M-\partial M$ (Hatcher's definition of a manifold with boundary being orientable is only for interior points).  There is some $\varepsilon$ such that $x$ is not in $\partial M\times (0,2\varepsilon)$, so $H_n(M,M-x;R)$ is naturally isomorphic to $H_n(M-\partial M,\partial M\times (0,\varepsilon);R)$, and it must be the local orientation at $x$ since $H_n(M,M-x;R)$ is naturally isomorphic to $H_n(U,U-x;R)$ for some open $U\subset M-\partial M\times(0,\varepsilon)$.
Edit: I've made a mess of further explanation by avoiding typing a commutative diagram, so I've at least drawn one.  The lemma gives a class $\alpha$ whose image in $H_n(M,\partial M)$ is something that, when restricted to a small enough neighborhood $U$, gives a local orientation.

