Steenrod Homology of a metric (or uniform, or Hausdorff, or …) space measures separations/holes by asking which discrete or approximate simplices can be filled-in to arbitrary fineness.
The dimension-$p$ simplices of a simplicial complex $K$ will be denoted $K_p$, and we shall write $A \ll X$ to mean that $A$ is a compact subspace of $X$.
Given a metric space $(X,d)$, define the category $Reg (X,d)$, “of regular complexes”, with objects pairs $(K,f)$ where
and with morphisms from $(K,f)$ to $(L,g)$ being the inclusions of subcomplexes $h : K \to L$ making $g$ an extension of $f$:
Note that $Reg_d$ has finite pushouts.
The condition that $K$ be locally-finite means that arbitrary formal sums $\sum g_x x$ of $p$-simplices $x\in K_p$ have a well-defined boundary in the usual way:
mentions a given simplex only finitely-many times. Taking $g_x\in G$, for any abelian group $G$, this gives a functor $C_* : Reg(X,d) \to Ch$, valued in chain complexes.
Definition : The Steenrod Homology $H_p(X,d)$ is the homology of the chain complex $\colim_{K,f} C_*$ in degree $p+1$.
In pieces, this means that a Steenrod $p$-cycle may be represented by a triple $(K,f,\varphi)$ of a locally finite complex $K$, a $K_1$-regular map $K_0 \to X$, and a formal sum $\varphi$ of simplices of dimension $p+1$ of $K$ with $\partial \varphi = 0$. The group operations may as well be represented in terms of the chains $\varphi$ for a single $K$ and $f$.
What we have just called $H_p$, Steenrod himself notated $H^{p+1}$. It was early days yet.
A compactly-supported Alexander-Spanier Cocycle is a (compactly-supported) cochain $\chi : X^{p+1} \to G$ whose derivation $d \chi$ vanishes on tuples that are small enough. For such a cochain, the sums
are finite sums, and vanish for both coboundaries $\chi = d\omega$ and for boundaries $x = \partial y$ (because $d d = \partial \partial = 0$. Thus the pairings descend to
and all the reasonable variations you might consider.
Given a set $\{ X_i \}$ of metric spaces, their disjoint union can be given a metric in which separate summands are all distance $1$ from eachother. This metric has $Reg( * \sqcup \coprod X_i) = colim Reg(* \sqcup X_i)$ and therefore also
While $limsup d(\partial_0 x,\partial_1 x) = 0$ is a topological condition on compact metric spaces, it definitely is not so on noncompact spaces. For this reason, one frequently considers the compactly-supported (compactly-generated?) homology as well,
Let $\dots \to X_{i+1} \to X_i \to \dots$ be a tower of compact metric spaces; in generous versions of Set-Theory, its limit is again a compact metric space. In contrast to $colim$, one can reasonably say $Reg(lim_i X_i) = lim_i Reg(X_i)$; and moreover, since the morphisms of $Reg(X_i)$ are inclusions of subcomplexes, it follows that
the usual abstract nonsense relating to the Milnor sequence, then gives short exact sequences
N. E. Steenrod, Regular Cycles of Compact Metric Spaces, Annals of Mathematics, Second Series, Vol. 41, No. 4 (Oct., 1940), pp. 833-851 JSTOR
John Milnor, On axiomatic homology theory, Pacific J. Math. Volume 12, Number 1 (1962), 337-341 (Euclid)
Last revised on August 9, 2022 at 14:48:39. See the history of this page for a list of all contributions to it.