nLab Steenrod homology

Idea

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.

Notation

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 $X$.

The Definition

Given a metric space $(X,d)$, define the category $Reg (X,d)$, “of regular complexes”, with objects pairs $(K,f)$ where

• $K$ is a locally-finite [simplicial complex]
• $f : K_0 \to X$ satisfies $limsup_{e:K_1} d( \partial_0 e, \partial_1 e) = 0$

and with morphisms from $(K,f)$ to $(L,g)$ being the inclusions of subcomplexes $h : K \to L$ making $g$ an extension of $f$:

$$Reg_d((K,f),(L,g)) = \{ h : hom(K,L) | f = g \circ h \}$$

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:

$$\partial \sum g_x x = \sum (-1)^j g_x \partial_j x$$

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 $f$.

Historical note

What we have just called $H_p$, Steenrod himself notated $H^{p+1}$. It was early days yet.

Pairing with Alexander-Spanier Cohomology

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

$$\varphi \frown \chi = \sum_{x :K_{p+1}} n_x (d\chi)(x) = \sum_x n_x \chi(\partial x)$$

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

$$H_p( X,d ;\mathbb{Z}) \otimes H_c^p( X , G ) \to G$$

and all the reasonable variations you might consider.

Continuities

Colimits

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

$$H_p(* \sqcup \coprod X_i) \simeq \bigoplus H_p( * \sqcup X_i )$$

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,

$${}_c H_p(X) = \colim_{A \ll X } H_p(A) .$$

Limits

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

$$\colim_{Reg(lim X)} C_* \simeq lim_i \colim_{Reg(X_i)} C_*$$

the usual abstract nonsense relating to the Milnor sequence, then gives short exact sequences

$$0 \to lim_i^1 H_{p+1}(X_i) \to H_p (lim X) \to lim_i H_p( X_i ) \to 0$$

Related entries

• Steenrod-Sitnikov homology

References

• 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)