Morse homology is naturally isomorphic to singular homology

Fix a target pair and write $X^j$ for its unstable $j$-skeleton. By [F1], $Y_{>j}={\bigcup }_{\mathrm{ind}(q)>j}{W}^s(q)$ is the $(n-j-1)$-skeleton of the stable CW structure, hence is closed. Put $U_j=M\setminus Y_{>j}$, with $U_{-1}=\mathrm{\varnothing }$ and $U_n=M$. Then $U_j$ is open, $U_{j-1}\subset U_j$, and $U_j\setminus U_{j-1}$ is the finite disjoint union of the stable manifolds of index $j$. Each such stable manifold is closed embedded in $U_j$: its compactified stable disk has boundary only in the removed higher-index stable cells, and its interior is its usual smooth immersed Euclidean manifold. The disk characteristic map therefore identifies its intrinsic and subspace topologies there. Also $X^j\subset U_j$, because a nonconstant intersection $W^u(p)\cap W^s(q)$ requires $\mathrm{ind}(p)>\mathrm{ind}(q)$.

Orient the normal quotient of each index-$j$ stable cell by its unstable critical ray. For a zero-dimensional stable cell take $\tau =0$ and its critical normal frame. Otherwise, in an invariant-axis hyperbolic chart its stable axis has strictly decreasing norm. Choose a small stable sphere within that chart and a fixed positive buffer time. Outside the sphere, its unique forward hitting time is smooth by transversality, and hitting time plus buffer places the point inside the chart. In an inner annular collar the same signed hitting-time formula remains smooth and positive after adding the buffer; multiply it by a smooth cutoff which is one near the sphere and zero on a smaller stable disk. This defines a smooth finite time $\tau (x)\ge 0$ on the entire stable cell, zero near $q$, with ${\varphi }_{\tau (x)}x$ always in the stable chart. Transport the local constant unstable normal basis at that point backwards by the normal quotient of $D{\varphi }_{-\tau (x)}$. It is a smooth full-rank global normal frame, equal to the critical frame near $q$; no asymptotic rescaling or equality of eigenvalues is used. The stable cell is contractible by [F1]: its compactified disk interior is homeomorphic to an open Euclidean disk. Choose disjoint tubular neighbourhoods of these finitely many closed stable cells in $U_j$ using [F4], with variable fibre radius. For a compatible metric $d$ on the stable base let $r(x)\le 1$ be the supremum of the allowable fibre radii inside the tube domain and a prescribed disjoint open neighbourhood. This function is positive and lower semicontinuous, by compact smaller fibre balls and openness. Then $r_0(x)={\inf }_y(r(y)\mathrm{/}2+d(x,y))$ is positive, continuous and at most $r(x)\mathrm{/}2$: a positive local lower bound for $r$ and the distance bound outside that neighbourhood prove positivity. Use radius $r_0$ and precompose the tube map with the inverse of its normal-quotient derivative. Excision in [F3] identifies $H_{\mathrm{\ell }}(U_j,U_{j-1};\mathrm{\Lambda })$ with the direct sum of these tube pairs punctured along their zero sections. The frame and radius identify each with the product of its contractible stable base and $({\mathbb{R}}^j,{\mathbb{R}}^j\setminus \{0\})$, so its homology is $\mathrm{\Lambda }$ in degree $j$ and zero otherwise. Use the chosen normal-ray fibre class $v_q$ as generator. In rank zero this is the chosen signed point unit, including a negative zero-dimensional ray.

Now let $F={\mathrm{\Psi }}_{S,-S}$ be evolution for a regular continuation from a source pair to the target pair. Every point of a source unstable cell of index at most $j$ lies at time $-S$ on a negative-end trajectory from its critical centre. Its evolved point has a target forward critical limit. If that target critical index exceeded $j$, it would be a continuation solution of negative index difference, forbidden by [F5]. Thus $F(X^{-}{}^j)\subset U_j^{+}$, and similarly $F(X^{-}{}^{j-1})\subset U_{j-1}^{+}$. It is an actual filtration-preserving map from the source unstable CW filtration to the target open stable-complement filtration. The final map is the diffeomorphism $F:M\to M$, isotopic to the identity by its smooth partial-time evolution.

Inclusion of the unstable skeletal pair $(X^j,X^{j-1})$ into $(U_j,U_{j-1})$ sends its oriented generator $e_p$ to $v_p$. Indeed an index-$j$ unstable cell meets the index-$j$ stable cells only at its own critical centre, and the normal map there is the identity on the unstable Hessian space, with the same chosen ray. Every other same-index intersection is excluded by the positive index-drop argument of [F5]. In degree zero the chosen signed vertex class and normal signed point unit agree. Thus the inclusion is an isomorphism on every consecutive relative group by [F3] and step 2.1, with identity matrix in the critical bases. Naturality of the pair connecting maps makes these relative-group isomorphisms commute with the connecting-defined boundaries.

Record the finite-filtration comparison, so no extra naturality is assumed. For a finite filtration $V_{-1}=\mathrm{\varnothing }\subset V_0\subset \cdots \subset V_n=V$ with consecutive relative groups concentrated in degree $j$, set $C_j=H_j(V_j,V_{j-1})$ and $d_j=i_{j-1}{\mathrm{\partial }}_j$. Exactness gives ${\mathrm{\partial }}_{j-1}{i}_{j-1}=0$, hence $d_{j-1}{d}_j=0$. Pair exactness also gives $H_{\mathrm{\ell }}(V_j)=0$ for $\mathrm{\ell }>j$ by induction, and $i_j:H_j(V_j)\to C_j$ is injective with image $\ker d_j$, since $i_{j-1}$ is injective as well. The pair $(V_{j+1},V_j)$ identifies $\ker d_j\mathrm{/}\mathrm{im}{d}_{j+1}$ with $H_j(V_{j+1})$. Later inclusions preserve that degree because their relative groups vanish there, hence this is $H_j(V)$. For $j=0$ use $i_0$ as the identity; for the final degree there is no next relative group. Every map used is an inclusion, quotient or connector, so the comparison commutes with every filtration-preserving map. This is the finite part of the cellular proof in [F3], and applies to the open $U_j$ filtration as well. Step 3.1 and final inclusion $X^n=M=U_n$ therefore identify its comparison with the usual cellular–singular one, without choosing a retraction of $U_j$.

Compute its relative degree-$j$ matrix. An oriented source characteristic disk for $p$ meets the target index-$j$ stable cells precisely at the finitely many rigid continuation solutions $\mathcal{C}(p,q)$; its boundary avoids those stable cells by step 2.2. Excision at these finitely many interior preimages sends its relative fundamental class to the sum of its local orientation classes. On each small neighbourhood the map into the normal-derivative-normalized target tube is the unstable source-to-stable-normal map of [F5]; its local degree is exactly $\tau (u)$. Projection to each fibre generator $v_q$ therefore gives ${\sum }_{u\in \mathcal{C}(p,q)}\tau (u)$, or its mod-two count. This conclusion uses only local smooth coordinates on the interior unstable cell; an auxiliary topological disk parametrization contributes degree $+1$ because it agrees with the critical ray on a fixed inner cap. Hence the filtered map of step 2.2 has exactly the continuation matrix in the critical bases of step 3.1.

Apply the natural finite-filtration comparison of step 4.1 to this map. By step 4.2 its chain map is the continuation count, and by [F2] its source and target generator maps are the Morse–cellular comparisons. The final singular map is $F_{\ast }\text{\hspace{0.17em}⁣}=\mathrm{id}$ by the isotopy in step 2.2 and [F3]. Thus ${\theta }_{+}{\mathrm{\Phi }}_{\ast }={\theta }_{-}$ exactly, proving the missing continuation-versus-comparison compatibility. For a fixed pair the skeleta are the intrinsic unions of its unstable cells; different critical heights, charts, bottle parametrizations or tube choices do not change them or their oriented relative generators. The pair-sequence construction therefore gives the same $\theta$. Reversing a critical ray changes its Morse and relative cellular generators together, leaving the represented singular class unchanged. This proves auxiliary and orientation independence rather than merely existence of unrelated isomorphisms.

Choose any actual metric Morse--Smale representative for the canonical homology of [F6]. The CW comparison of [F1] applies directly to it. Transition from any other representative is canonical by [F6], and step 5.1 makes the resulting singular identification independent of the representative. For a field-version pair choose a metric realizing $g(X,\cdot )=-df$, as in the disk construction of [F1]. Two such realizations have the same complex: their convex metric interpolation still satisfies this identity and hence has the same autonomous equation. Its transverse endpoint sequence gives the identity count, with the same critical rays. Thus metric realization adds no ambiguity. Composition, inverse and datum independence in [F6] complete $H{M}_{\ast }(M;\mathrm{\Lambda })\cong H_{\ast }(M;\mathrm{\Lambda })$ with exactly the choice-independence meaning of naturality in the statement.