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-pp simplices of a simplicial complex KK will be noted K pK_p. And, today, I feel like writing AXA \ll X to say that AA is a compact subspace of XX.

The Definition

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

  • K K is a locally-finite [simplicial complex]
  • f:K 0X f : K_0 \to X satisfies limsup e:K 1d( 0e, 1e)=0 limsup_{e:K_1} d( \partial_0 e, \partial_1 e) = 0

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

(1)Reg d((K,f),(L,g))={h:hom(K,L)|f=gh} Reg_d((K,f),(L,g)) = \{ h : hom(K,L) | f = g \circ h \}

Note that Reg dReg_d has finite pushouts.

The condition that KK be locally-finite means that arbitrary formal sums g xx\sum g_x x of pp-simplices xK px\in K_p have a well-defined boundary in the usual way:

(2)g xx=(1) jg x jx \partial \sum g_x x = \sum (-1)^j g_x \partial_j x

mentions a given simplex only finitely-many times. Taking g xGg_x\in G, for any abelian group GG, this gives a functor C *:Reg(X,d)ChC_* : Reg(X,d) \to Ch , valued in chain complexes.

Definition : The Steenrod Homology H p(X,d)H_p(X,d) is the homology of the chain complex colim K,fC *\colim_{K,f} C_* in degree p+1p+1.

In pieces, this means that a Steenrod pp-cycle may be represented by a 3ple (K,f,φ)(K,f,\varphi) of a locally-finite complex KK, a K 1K_1-regular map K 0XK_0 \to X, and a formal sum φ\varphi of simplices of dimension p+1p+1 of KK with φ=0\partial \varphi = 0. The group operations may as well be represented in terms of the chains φ\varphi for a single KK and ff.

Historical note

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

Pairing with Alexander-Spanier Cohomology

A compactly-supported Alexander-Spanier Cocycle is a (compactly-supported) cochain χ:X p+1G\chi : X^{p+1} \to G whose derivation dχd \chi vanishes on tuples that are small enough. For such a cochain, the sums

(3)φχ= x:K p+1n x(dχ)(x)= xn xχ(x) \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 χ=dω\chi = d\omega and for boundaries x=yx = \partial y (because dd==0d d = \partial \partial = 0. Thus the pairings descend to

(4)H p(X,d;)H c p(X,G)G 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}\{ X_i \} of metric spaces, their disjoint union can be given a metric in which separate summands are all distance 11 from eachother. This metric has Reg(*X i)=colimReg(*X i) Reg( * \sqcup \coprod X_i) = colim Reg(* \sqcup X_i) and therefore also

(5)H p(*X i)H p(*X i) H_p(* \sqcup \coprod X_i) \simeq \bigoplus H_p( * \sqcup X_i )

While limsupd( 0x, 1x)=0limsup 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,

(6) cH p(X)=colim AXH p(A). {}_c H_p(X) = \colim_{A \ll X } H_p(A) .

Limits

Let X i+1X i\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 colimcolim, one can reasonably say Reg(lim iX i)=lim iReg(X i)Reg(lim_i X_i) = lim_i Reg(X_i); and moreover, since the morphisms of Reg(X i)Reg(X_i) are inclusions of subcomplexes, it follows that

(7)colim Reg(limX)C *lim icolim Reg(X i)C * \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

(8)0lim i 1H p+1(X i)H p(limX)lim iH p(X i)0 0 \to lim_i^1 H_{p+1}(X_i) \to H_p (lim X) \to lim_i H_p( X_i ) \to 0

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)

Last revised on January 30, 2018 at 02:52:15. See the history of this page for a list of all contributions to it.