# Eilenberg–Steenrod axioms for homology without pairs of spaces

Say a functor $H\colon \mathrm{Top} → \mathrm{Ab}^ℤ$ satisfies the following set of axioms:

1. Homotopy: If maps $f \colon X → Y$ and $g\colon X → Y$ are homotopic, then $H(f) = H(g)$.

2. Excision’: If $T ⊂ X$ and $S ⊂ X$ are subsets of a space $X$ such that $\operatorname{cls} S ⊂ \operatorname{int} T$, then the inclusion $ι \colon (X\setminus S) /\!/ (T\setminus S) → X /\!/ T$ induces an isomorphism $H(ι)$.¹

3. Dimension: For any one-point space $\star$, $H(\star) = 0$.

4. Additivity: $H$ preserves all small coproducts.

5. Exactness’: If $T ⊂ X$ is a subset of a space $X$, then there’s a long exact sequence of graded abelian groups $$H[+1] (X /\!/ T) \overset{δ[+1]}→ H(T) \overset{H(ι)}→ H(X) \overset{H(π)}→ H(X /\!/ T) \overset{δ}→ H[-1](T),$$ where $ι\colon T → X$ and $π \colon X → X /\!/ T$ are inclusions and $δ\colon H∘(–/\!/•) → H[-1]∘(•)$ is a natural transformation of functors $\mathrm{Top}^2 → \mathrm{Ab}^ℤ$.²³

Is $H$ then a homology theory? (Maybe in the sense that if $H' \colon \mathrm{Top}^2 → \mathrm{Ab}^ℤ$ satisfies the usual Eilenberg–Steenrod axioms, then $\mathrm{Top} \overset{(–,∅)}→ \mathrm{Top}^2 \overset{H'}→ \mathrm{Ab}^ℤ$ satisfies these modified versions as well.)

¹,²:Here, for $A ⊂ X$, the “double quotient of $X$ by $A$” is defined as $X/\!/A := \operatorname{cone} (A → X)$.

³: Here, for $k ∈ ℤ$, $H[k]$ denotes $H$ postcomponed by the endofunctor $\mathrm{Ab}^ℤ → \mathrm{Ab}^ℤ$ which shifts the degree by $k$ (so e.g. $H_n[+1] = H_{n+1}$).