Cw Complexes and Cellular Homology

The defining equivalence relation identifies only boundary points of $D^n$ with points of $X$ (and may identify boundary points with one another); no point of ${\mathring{D}}^n$ is equivalent to a distinct point.

The saturated open set ${\mathring{D}}^n$ has quotient homeomorphic to its image, and its complement in the closed-cell image is the image of $S^{n-1}$. Thus the image is open relative to the attached space, while step 1.1 deliberately says nothing about the boundary.

Let ${\chi }_e:D^m\to X$ be the characteristic map of an $m$-cell. If $m\le n$, then ${\chi }_e^{-1}(X^n)=D^m$. If $m>n$, this inverse image lies in $S^{m-1}$ and equals the inverse image there of the already closed lower skeleton $X^n\subseteq X^{m-1}$; induction on $m$ makes it closed in $S^{m-1}$ and hence in $D^m$. The weak-topology clause now makes $X^n$ closed in $X$.

The open-cell interiors are disjoint by the cellular partition, and The interior of an attached cell embeds openly in its closure identifies precisely the new interior points after $X^{n-1}$. This gives the asserted difference.

If $K$ met infinitely many cells, choose $x_i\in K$ in pairwise distinct cells and put $S=\{x_i:i\ge 1\}$. Induction over the skeleta shows that $S\cap X^n$ is closed in $X^n$: on the boundary of each characteristic $n$-disk this follows from the induction hypothesis, and the disk interior contains at most one $x_i$. The weak topology then closes the induction.

The same argument applies to every subset of $S$, so $S$ is a closed discrete subspace of $K$. An infinite discrete space is not compact, contradicting compactness of the closed subspace $S\subseteq K$.

The image is compact, so A compact subspace of a CW complex meets only finitely many cells gives finitely many cells meeting it.

Add the finitely many cells in the closures of those cells, repeating down dimensions. Closure finiteness makes this terminate in a finite union, and the result is a CW subcomplex by Skeleta, CW subcomplexes, and relative CW complexes.

On each attached disk, the already-defined map on $S^{n-1}\times I\cup D^n\times \{0\}$ extends over $D^n\times I$ because this subspace is a deformation retract of a collar-containing neighborhood in the disk cylinder.

Induct over attachment stages, choosing these extensions; their restrictions agree on prior stages. The weak topology of the relative CW construction makes the assembled map $X\times I\to Y$ continuous and it extends $H$.

Necessity follows by restriction. Conversely, $f{\mathrm{\mid }}_{\bar{e}}$ is continuous because $\bar{e}$ lies in the skeleton of its dimension.

For closed $C\subseteq Y$, each $f^{-1}(C)\cap \bar{e}$ is closed in $\bar{e}$ by step 1.1; the weak-topology clause in CW complex with closure finiteness and weak topology makes $f^{-1}(C)$ closed.

For $n\ge 1$, the pair $(X^n,X^{n-1})$ is a good pair (collar neighborhoods in the attached disks retract to $X^{n-1}$), and collapsing $X^{n-1}$ gives the wedge ${\bigvee }_{\alpha }{D}^n\mathrm{/}{S}^{n-1}\cong {\bigvee }_{\alpha }{S}^n$. For $n=0$, $X^0$ is discrete and the same direct-sum calculation follows directly from its singular chain complex.

If there are no $n$-cells, both sides are zero. Otherwise, for $n\ge 1$, Good pairs and quotient reduced homology and Homology of spheres give one copy of $G$ in degree $n$ and zero otherwise. A singular simplex has compact image, so A compact subspace of a CW complex meets only finitely many cells makes every chain involve only finitely many wedge summands; the wedge group is therefore the direct sum, not a product. The direct $n=0$ calculation in step 1.1 gives the same conclusion.

For $n=1$ the composite is zero because $d_0=0$ by definition.

Let $n\ge 2$. Write $d_n=j_{n-1}{\mathrm{\partial }}_n,d_{n-1}=j_{n-2}{\mathrm{\partial }}_{n-1},$ where $j_{n-1}:H_{n-1}(X^{n-1})\to H_{n-1}(X^{n-1},X^{n-2})$ is the quotient map. In the long exact sequence of the pair $(X^{n-1},X^{n-2})$, $j_{n-1}$ is immediately followed by ${\mathrm{\partial }}_{n-1}$.

Exactness gives ${\mathrm{\partial }}_{n-1}{j}_{n-1}=0$, and hence $d_{n-1}{d}_n=j_{n-2}({\mathrm{\partial }}_{n-1}{j}_{n-1}){\mathrm{\partial }}_n=0.$ Together with step 1.1 this covers every $n\ge 1$.

Its relative homology vanishes outside degree $n+1$ by Relative homology of consecutive CW skeleta.

In the pair long exact sequence, the two relative groups adjacent to $H_k(X^n)$ vanish for $k<n$, and only the outgoing one need vanish for $k=n$; exactness gives respectively an isomorphism and a surjection.

A singular chain has finitely many simplices, so its image is compact; The image of a compact space lies in a finite CW subcomplex places it in one finite subcomplex and hence in some skeleton. Thus every homology class is in the image.

If a class from $H_k(X^n)$ dies in $H_k(X)$, a finite singular bounding chain has compact image and lies in some $X^m$. It already kills the class there, proving injectivity of the colimit map.

For every finite $m\ge 0$, $H_k(X^m)=0$ when $k>m$: begin with the discrete $X^0$ case of [F1], and use [F2] successively with [F1] to pass from $X^{m-1}$ to $X^m$. The empty skeleton also has zero homology.

For $n\ge 1$, the exact sequence of $(X^n,X^{n-1})$ and step 1.1 give an injection $i_n:H_n(X^n)\to C_n$ with image $\ker {\mathrm{\partial }}_n$. The same argument one degree lower makes $i_{n-1}$ injective, including $i_0:H_0(X^0)\to H_0(X^0,\mathrm{\varnothing })$, which is an isomorphism. Since $d_n=i_{n-1}{\mathrm{\partial }}_n$, we obtain $\ker d_n=\ker {\mathrm{\partial }}_n=i_n{H}_n(X^n)$. For $n=0$ this identity follows directly from $i_0$ being an isomorphism and $d_0=0$.

For $(X^{n+1},X^n)$, exactness and $H_n(X^{n+1},X^n)=0$ give the exact tail $C_{n+1}\stackrel{{\mathrm{\partial }}_{n+1}}{\to }{H}_n(X^n)⟶H_n(X^{n+1})⟶0.$ Under the injection $i_n$ of step 2.1, $\mathrm{im}{\mathrm{\partial }}_{n+1}$ corresponds exactly to $\mathrm{im}{d}_{n+1}$ by [F5]. Taking the quotient therefore yields $H_n^{\mathrm{c}\mathrm{e}\mathrm{l}\mathrm{l}}(X;G)\cong H_n(X^n)\mathrm{/}\mathrm{im}{\mathrm{\partial }}_{n+1}\cong H_n(X^{n+1};G)\mathrm{.}$

By [F3], all inclusions after $X^{n+1}$ induce isomorphisms on $H_n$. By [F4], their colimit is $H_n(X;G)$. Combining with step 3.1 proves the comparison for arbitrary, possibly infinite-dimensional, CW complexes.

A cellular map preserves all skeleta. Its induced chain maps commute with inclusions, quotient maps and connecting homomorphisms: the latter send a relative cycle represented by $c$ to the class of $\mathrm{\partial }c$, and a chain map commutes with $\mathrm{\partial }$. Consequently every injection, quotient identification and colimit map in steps 2.1--4.1 commutes with cellular maps. The comparison is therefore natural in exactly the stated sense.

If $A=\mathrm{\varnothing }$, the assertion is the absolute cellular-homology theorem. Suppose $A\mathrm{\ne }\mathrm{\varnothing }$. The quotient $X\mathrm{/}A$ is a CW complex with one base vertex coming from $A$ and exactly the cells of $X\setminus A$ otherwise. Consequently its reduced cellular complex is canonically $C_{\ast }^{\mathrm{c}\mathrm{e}\mathrm{l}\mathrm{l}}(X;G)\mathrm{/}{C}_{\ast }^{\mathrm{c}\mathrm{e}\mathrm{l}\mathrm{l}}(A;G)$, including in degree zero.

A CW subcomplex is closed. The cellwise radial collar construction, assembled over the skeleta by the weak topology, gives an open neighborhood $V$ of $A$ that deformation retracts onto $A$ while fixing $A$; equivalently, every nonempty CW pair is a good pair. Therefore Good pairs and quotient reduced homology identifies $H_{\ast }(X,A;G)$ with ${\tilde{H}}_{\ast }(X\mathrm{/}A;G)$. Applying Cellular homology computes singular homology to $X\mathrm{/}A$ and using step 1.1 proves the claim. Under this identification, the triple connecting maps for the relative skeleta are exactly the quotient cellular differential.

Project the three-skeleton connecting map defining $d_n$ to the summand of a fixed $(n-1)$-cell.

For $n\ge 2$, first pass from $X^{n-1}$ to $X^{n-1}\mathrm{/}{X}^{n-2}$ as required by the relative target of the cellular boundary, and then collapse all summand spheres except that of $e_{\beta }^{n-1}$. Equivalently, collapse the entire complement $X^{n-1}\setminus e_{\beta }^{n-1}$, including $X^{n-2}$. Naturality of the connecting homomorphism for the characteristic disk identifies the coefficient with the composite from its oriented boundary sphere to this quotient sphere. This is precisely the reduced-homology endomorphism in Incidence number of two CW cells, so its degree is $[e_{\alpha }^n:e_{\beta }^{n-1}]$.

For $n=1$, the connecting homomorphism sends an oriented characteristic interval to its terminal endpoint minus its initial endpoint, which is exactly the separately defined incidence number. The boundary is an element of a direct sum, so only finitely many coefficients are nonzero; equality of all its projections therefore proves the formula for every $n\ge 1$; separately, $d_0=0$ by definition.

The restrictions $f:(X^n,X^{n-1})\to (Y^n,Y^{n-1})$ induce maps of relative groups, hence maps on cellular chains.

Naturality of pair connecting maps makes these maps commute with the differentials. Under Cellular homology computes singular homology, they are induced by the same maps of pairs, hence agree with $f_{\ast }$ on singular homology.

For every $n$, either $C_n^{\mathrm{c}\mathrm{e}\mathrm{l}\mathrm{l}}$ or $C_{n-1}^{\mathrm{c}\mathrm{e}\mathrm{l}\mathrm{l}}$ is zero, so $d_n=0$.

Thus cellular homology equals the chain group in each degree; Cellular homology computes singular homology transfers this calculation to singular homology.

Rank-nullity in each degree gives $\mathrm{rank}{C}_n=\mathrm{rank}{H}_n^{\mathrm{c}\mathrm{e}\mathrm{l}\mathrm{l}}+\mathrm{rank}\mathrm{im}{d}_n+\mathrm{rank}\mathrm{im}{d}_{n+1}$.

Alternating and summing cancels each boundary rank with its neighboring occurrence. The left side is Euler characteristic of a finite CW complex, and Cellular homology computes singular homology identifies the remaining ranks with singular homology.

The cells of $X$ are the disjoint union of the cells of $A$ and those outside $A$, so the first equality follows by splitting the finite alternating sum.

If $A$ is nonempty, $X\mathrm{/}A$ has one new $0$-cell (the quotient point) and exactly the cells outside $A$ otherwise; its alternating count gives the second formula.

Product cells have dimension $p+q$, and there are $c_p(X)c_q(Y)$ of them in that dimension.

The finite alternating double sum is ${\sum }_{p,q}(-1{)}^{p+q}{c}_p(X)c_q(Y)=({\sum }_p(-1{)}^p{c}_p(X))({\sum }_q(-1{)}^q{c}_q(Y))$, which is the required formula.