The rank of an abelian group $A$, defined as $\dim_{\mathbb{Q}} (A \otimes_{\mathbb{Z}} \mathbb{Q})$.
For a finitely generated abelian group $A \cong \mathbb{Z}^r \oplus T$ (with $T$ torsion), this returns $r$, the number of $\mathbb{Z}$-summands.
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Lemma 18.2 (rank is additive on short exact sequences). Given a SES $0 \to A \to B \to C \to 0$ of finitely generated abelian groups, $\operatorname{rank} B = \operatorname{rank} A + \operatorname{rank} C$.
Proof: Tensoring with $\mathbb{Q}$ over $\mathbb{Z}$ is exact (since $\mathbb{Q}$ is $\mathbb{Z}$-flat), so the rationalized sequence is also short exact, and the result follows from rank-nullity for $\mathbb{Q}$-vector spaces.
The rank of an abelian group is an isomorphism invariant: if $A \cong B$ as abelian groups, then $\operatorname{rank} A = \operatorname{rank} B$.
The number $a_n$ of $n$-cells in a CW-structure on $X$, i.e. the cardinality of the indexing set of $n$-dimensional cells.
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The Euler characteristic of a CW-complex $X$ of dimension $< N$, defined as the alternating sum $$\chi(X) = \sum_{k=0}^{N-1} (-1)^k a_k$$ where $a_k$ is the number of $k$-cells.
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The $n$th Betti number of $X$: the $\mathbb{Z}$-rank of the singular homology group $H_n(X; \mathbb{Z})$.
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Homotopy-invariance of Betti numbers: a homotopy equivalence $X \simeq Y$ induces $\operatorname{rank} H_k(X) = \operatorname{rank} H_k(Y)$ for every $k$.
Telescoping identity for alternating sums: for any sequence $b$, $$\sum_{k=0}^{N-1} (-1)^k b_k + \sum_{k=0}^{N-1} (-1)^k b_{k+1} = b_0 - (-1)^N b_N.$$ Used in the proof that $\chi(X)$ equals the alternating sum of Betti numbers.
Algebraic backbone of Theorem 18.3: if a chain sequence $c_k$ decomposes as $c_k = h_k + b_k + b_{k+1}$ with $b_0 = b_N = 0$, then the alternating sums $\sum (-1)^k c_k$ and $\sum (-1)^k h_k$ agree. Applied with $c_k$ = number of $k$-cells, $h_k$ = $k$th Betti number, $b_k$ = rank of $k$-boundaries.
The rank of the free abelian group on a finite set $S$ equals $|S|$.
The rank of cellular homology equals the rank of singular homology, via the isomorphism $H_k^{\text{cell}}(X) \cong H_k(X)$ for CW-complexes (Theorem 16.3).
The differential $d_{k+1} : C_{k+1} \to C_k$ of a chain complex of abelian groups, packaged as a $\mathbb{Z}$-linear map.
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Rank-nullity for $\mathbb{Z}$-linear maps between finitely generated abelian groups: $\operatorname{rank} A = \operatorname{rank}(\ker f) + \operatorname{rank}(\operatorname{im} f)$.
Rank decomposition of cycles at degree $k+1$: for a chain complex with finitely generated pieces, $$\operatorname{rank} Z_k = \operatorname{rank} B_{k+1} + \operatorname{rank} H_{k+1}$$ i.e. the cycles in degree $k$ split via the short exact sequence $0 \to B_{k+1} \to Z_k \to H_{k+1} \to 0$.
Rank decomposition at degree $0$: since there is no differential into $C_0$, all of $C_0$ consists of $0$-cycles, giving $\operatorname{rank} C_0 = \operatorname{rank} B_0 + \operatorname{rank} H_0$.
The trivial submodule $\{0\}$ has rank $0$.
Combined rank decomposition of a bounded chain complex of finitely generated abelian groups: there exist sequences $z_k = \operatorname{rank} Z_k$ and $b_k = \operatorname{rank} B_k$ satisfying $\operatorname{rank} C_k = z_k + b_{k-1}$ and $z_k = b_k + \operatorname{rank} H_k$, with $b_k = 0$ above the dimension bound. This provides the structural identity used to prove $\chi(X) = \sum (-1)^k \operatorname{rank} H_k(X)$.
If $X$ has finitely many cells in each dimension, then each cellular chain group $C_k(X)$ is finitely generated as a $\mathbb{Z}$-module.
If $X$ has no cells in dimensions $\ge N$, then the cellular chain group $C_m(X)$ is zero for $m \ge N$.
The rank of the cellular $k$-chain group equals the number $a_k$ of $k$-cells, since $C_k(X) = \mathbb{Z}[A_k]$ is free abelian on the indexing set of $k$-cells.
Specialization of chain_complex_rank_decomposition to the cellular chain
complex of a finite-dimensional CW-complex: the number of $k$-cells decomposes
as $a_k = z_k + b_{k-1}$ and $z_k = b_k + \operatorname{rank} H_k(X)$.
Repackaged form of cellular_chain_complex_ranks: there is a sequence
$b : \mathbb{N} \to \mathbb{Z}$ vanishing at $0$ and at $N$ such that for each
$k < N$,
$$a_k = \operatorname{rank} H_k(X) + b_k + b_{k+1}.$$
This is the input to the telescoping argument behind Theorem 18.3.
Theorem 18.3. For a finite CW-complex $X$, $$\chi(X) = \sum_{k} (-1)^k \operatorname{rank} H_k(X).$$ That is, the Euler characteristic computed from cell counts coincides with the alternating sum of Betti numbers.
Theorem 18.1. The Euler characteristic $\chi(X) = \sum (-1)^k a_k$ of a finite CW-complex depends only on the homotopy type of $X$, not on the choice of CW-structure. Combining Theorem 18.3 with homotopy-invariance of singular homology.
Künneth-style identity for Betti numbers under products: $$\sum_n (-1)^n b_n(X \times Y) = \Bigl(\sum_p (-1)^p b_p(X)\Bigr) \Bigl(\sum_q (-1)^q b_q(Y)\Bigr),$$ expressed as a double sum. This is the rank-level shadow of the Künneth formula and underlies multiplicativity $\chi(X \times Y) = \chi(X) \chi(Y)$.
The number of torsion coefficients of an abelian group $A$: the minimum number of generators needed for its torsion subgroup. For a finitely generated abelian group $A \cong \mathbb{Z}^r \oplus \bigoplus_i \mathbb{Z}/d_i$ this is the number of cyclic torsion summands.
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The minimal number of $k$-cells predicted by Wall's theorem: $r(k) + t(k) + t(k - 1)$, where $r(k) = \operatorname{rank} H_k(X)$ is the $k$th Betti number and $t(k)$ is the number of torsion coefficients of $H_k(X)$. (For $k = 0$ the term $t(k - 1)$ is omitted.)
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Theorem 18.4 (Wall's theorem). Let $X$ be a simply connected CW-complex of finite type. Then there exists a CW-complex $Y$ with exactly $r(k) + t(k) + t(k - 1)$ cells in dimension $k$, together with a homotopy equivalence $Y \to X$.
A Moore space $M(A, k)$ is a path-connected CW-complex whose only nontrivial reduced homology group is $H_k(M) \cong A$.
- pathConnected : PathConnectedSpace M
- homology_iso : Nonempty (((AlgebraicTopology.singularHomologyFunctor AddCommGrpCat k).obj (AddCommGrpCat.of ℤ)).obj (TopCat.of M) ≅ A)
- homology_vanish (q : ℕ) : 0 < q → q ≠ k → CategoryTheory.Limits.IsZero (((AlgebraicTopology.singularHomologyFunctor AddCommGrpCat q).obj (AddCommGrpCat.of ℤ)).obj (TopCat.of M))
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Construction step for Moore spaces: for any abelian group $A$ and any $k > 0$, there exists a path-connected CW-complex $M$ whose cellular short complex at level $k$ has homology $A$ and vanishes elsewhere.
Translation of moore_space_skeleton_SC_computation into cellular
homology: for any abelian group $A$ and any $k > 0$, there is a
path-connected CW-complex $M$ with cellular homology $A$ in degree $k$ and
$0$ in all other positive degrees.
Existence of Moore spaces. For any abelian group $A$ and any $k > 0$, there exists a Moore space $M(A, k)$ — a path-connected CW-complex with $H_k(M) \cong A$ and $H_q(M) = 0$ for $q > 0, q \neq k$. Used as a building block in Proposition 18.5 to realize arbitrary graded abelian groups as reduced homology of a CW-complex.
Wedge construction (data version): given a family of path-connected CW spaces $M(k)$ for each $k > 0$ whose only nonzero positive homology is in degree $k$, there exists a path-connected CW-complex $X$ together with isomorphisms $H_k(M(k)) \cong H_k(X)$ in each positive degree.
Wedge construction (homology version, symmetric reformulation of
wedge_CW_construction_data): the wedged CW-complex $X$ has its homology
groups isomorphic to the corresponding homology groups of the individual
Moore spaces $M(k)$.
Proposition 18.5. Realization of graded abelian groups as reduced homology: for any graded abelian group $G_*$ with $G_k = 0$ for $k \le 0$, there exists a path-connected CW-complex $X$ with $\widetilde H_*(X) \cong G_*$.
Proof: build a Moore space $M(G_k, k)$ for each $k > 0$ and wedge them together.