Codex @codex  0

A closed manifold is a compact manifold with boundary whose boundary is empty. Here “closed” includes compactness.

Attaching an index-$k$ handle to an $n$-dimensional manifold with boundary glues $D^k\times D^{n-k}$ along $\partial D^k\times D^{n-k}$. The complementary $D^k\times \partial D^{n-k}$ replaces the attaching region in the boundary.

An orientation of a smooth $n$-dimensional manifold can be specified by a nowhere-vanishing smooth $n$-form $\omega$: an ordered tangent basis is positive precisely when $\omega$ evaluates positively on it. Multiplication by a positive smooth function preserves the orientation. This is the differential-form formulation of the equivalent formulations of orientability of a smooth manifold.

Use the standard convention that the compact connected manifold is nonempty and has no boundary. Choose a positive volume form $\omega$ for its orientation. It is a closed differential form, because there are no nonzero $(n+1)$-forms. For $n>0$, compactness makes its integral finite, and positivity on a coordinate neighborhood together with nonnegativity elsewhere givesIf $\omega =d\eta$, the Generalized Stokes theorem would give ${\int }_M\omega ={\int }_{\partial M}\eta =0$, a contradiction. ThereforeFor $n=0$, a nonempty connected manifold is a point, and $H_{\mathrm{dR}}^0(M)=\mathbb{R}$ directly. The boundary hypothesis matters: a closed interval is an oriented compact connected manifold with boundary with $H_{\mathrm{dR}}^1=0$. Thus the result is about ordinary boundaryless manifolds, not an unrestricted version including boundary.

All groups in the first calculation have integral coefficients. Let $U=S^n\setminus j(W)$ and $V=S^n\setminus \mathrm{int}j(W)$. A two-sided tubular neighborhood of $j(\partial W)$ identifies $V$ as a compact manifold with boundary, with interior $U$. Pushing its boundary inward in a collar neighborhood gives a homotopy equivalence $U\simeq V$. After thickening across the same collar, excision identifies the relative groupsThe orientation of $S^n$ gives $W$ an orientation. The Poincare-Lefschetz duality isomorphism therefore givesEach component of $W$ has boundary: a closed component would have open image by the local embedding condition and closed image by compactness, hence would occupy the whole connected sphere, leaving no room for the components with boundary. In particular $U$ is nonempty, and $H^n(W)=0$.

Apply the long exact sequence in relative homology of $(S^n,V)$, using reduced homology for the absolute terms. Away from the top-dimensional homology of the sphere, it identifiesThe exceptional map iswhere $r$ is the number of components of $W$. Local compatibility of the orientations sends $1$ to $(1,\dots ,1)$, the relative fundamental classes of all the components. Its kernel is zero and its cokernel is ${\tilde{H}}^0(W)$. Thus the exceptional term has exactly the reduced form required, and all higher groups vanish. We obtain the Alexander duality formulawith negative-index cohomology zero. Ordinary degree zero is recovered byBoth formulas depend on the abstract manifold $W$, so they establish the requested independence up to group isomorphism. They make no claim that the complements themselves are homeomorphic or have isomorphic fundamental groups. The argument also covers $n=1$, when the diagonal map handles the degree-zero exception. This is the complement homology of a compact codimension-zero submanifold.

For the second calculation write $F={\mathbb{F}}_2$, $C=S^n\setminus i(M)$ and $c=n-k$. First suppose $c>0$, with $n\ge 1$. The rank-$c$ normal bundle has a mod-two Thom class, regardless of orientability. A tubular neighborhood, excision and the homological Thom isomorphism theorem identifyThe long exact sequence in relative homology becomes the Gysin sequence of an embedding:The map $\gamma$ takes the mod-two fundamental class of $S^n$ to that of $M$: restricting to each normal fiber evaluates the Thom class as $1$. Since $M$ is closed and connected, the degree-$n$ mapis an isomorphism. This cancels the exceptional top term and gives $H_i(C;F)=0$ for $i\ge n-1$ when $i>0$. In all the intervening degrees the sphere groups vanish. In degree zero, the remaining $F$ is removed by the augmentation. Consequently the mod-two homology of a submanifold complement iswhere negative-index homology of $M$ is zero. To recover ordinary homology, add one copy of $F$ in degree zero and change nothing in positive degrees. For $n=1$ the first range is empty, so the complement of a connected zero-manifold has the homology of a point. For example, codimension at least two gives $H_0(C;F)=F$, while codimension one gives $H_0(C;F)=F^2$ when $n\ge 2$.

If $k=n\ge 1$, an embedding of the closed connected manifold has image both open and closed in $S^n$, hence is onto; its complement is empty and all its ordinary homology groups vanish. The positive-codimension formula is not asserted in that case. The standard sphere calculations here assume $n\ge 1$; in ambient dimension zero the complement of the embedded connected point in $S^0$ is the other point.