View source for Steenrod-Sitnikov homology

A [[Homology theory|homology theory]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120240/s1202401.png" /> supposed to be defined on the [[Category|category]] of pairs of compact metric (i.e., metrizable) spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120240/s1202402.png" />, satisfying all [[Steenrod–Eilenberg axioms|Steenrod–Eilenberg axioms]] (in the case of generalized Steenrod–Sitnikov homology, without a dimension axiom) together with a strong excision axiom (i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120240/s1202403.png" /> induces an isomorphism in homology).

For an ordinary Steenrod–Sitnikov homology theory <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120240/s1202404.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120240/s1202405.png" /> an Abelian group) J. Milnor [[#References|[a3]]] established the following axiomatic characterization: An ordinary homology theory <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120240/s1202406.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120240/s1202407.png" /> satisfying the Eilenberg–Steenrod axioms (with strong excision and dimension axiom) and in addition the cluster (or strong wedge) axiom is isomorphic (as a homology theory) to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120240/s1202408.png" />.

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120240/s1202409.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120240/s12024010.png" />, be a family of based spaces in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120240/s12024011.png" />; then the cluster <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120240/s12024012.png" /> is the wedge of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120240/s12024013.png" />, equipped with the [[Strong topology|strong topology]] (a neighbourhood of the basepoint <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120240/s12024014.png" /> contains almost all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120240/s12024015.png" /> or, alternatively, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120240/s12024016.png" />). The projection <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120240/s12024017.png" /> induces a mapping

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120240/s12024018.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a1)</td></tr></table>

The cluster axiom requires that (a1) be an isomorphism.

The cluster axiom turns out to be some kind of continuity axiom: A homology theory <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120240/s12024019.png" /> is continuous whenever there is a natural isomorphism

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120240/s12024020.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a2)</td></tr></table>

i.e. the functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120240/s12024021.png" /> commutes with inverse limits. S. Eilenberg and N. Steenrod [[#References|[a2]]] discovered that there is a conflict between (a2) and the exactness axiom, so that no homology theory can be continuous. However, the cluster axiom reveals itself as a special case of (a2): One has

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and

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Milnor's uniqueness theorem admits an extension for generalized homology theories [[#References|[a1]]]: A homology theory <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120240/s12024024.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120240/s12024025.png" /> satisfying the Milnor axioms (now without a dimension axiom) is, up to an isomorphism of homology theories, uniquely determined by its restriction to the category of compact absolute neighbourhood retracts <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120240/s12024026.png" /> (or polyhedra or CW-spaces). Moreover, every homology theory <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120240/s12024027.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120240/s12024028.png" /> admits a unique extension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120240/s12024029.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120240/s12024030.png" /> as a Steenrod–Sitnikov homology theory. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120240/s12024031.png" /> (the homology theory with coefficients in a spectrum, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120240/s12024032.png" />) (cf. also [[Spectrum of spaces|Spectrum of spaces]]), then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120240/s12024033.png" /> is uniquely determined by the spectrum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120240/s12024034.png" />.

It turns out [[#References|[a1]]] that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120240/s12024035.png" /> is constructed like <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120240/s12024036.png" />, but with replacing continuous mappings by strong shape mappings. So, Steenrod–Sitnikov homology is defined on a strong shape category (cf. [[Strong shape theory|Strong shape theory]]).

Steenrod–Sitnikov homology appears as the appropriate tool for handling geometric problems in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120240/s12024037.png" />. This pertains to [[Alexander duality|Alexander duality]] in its most modern form, dealing with homology and cohomology with coefficients in a spectrum, to [[S-duality|<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120240/s12024038.png" />-duality]], as well as to dimension theory of subspaces of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120240/s12024039.png" />.

Ordinary Steenrod–Sitnikov homology appeared for the first time in [[#References|[a6]]] as a tool for Alexander duality. In the 1950s, K. Sitnikov rediscovered <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120240/s12024040.png" /> by using a different definition [[#References|[a4]]], [[#References|[a5]]], without knowing about [[#References|[a6]]], and verified an Alexander duality theorem for arbitrary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120240/s12024041.png" /> by using this kind of homology with compact support.

As official opponent in Sitnikov's doctoral dissertation, G.S. Chogoshvili pointed out that the homology groups of Sitnikov and Steenrod agree.

Sitnikov's definition of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120240/s12024042.png" /> is modelled after the classical definition of [[Vietoris homology|Vietoris homology]]: A Vietoris cycle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120240/s12024043.png" /> on a space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120240/s12024044.png" /> consists of a sequence of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120240/s12024046.png" />-cycles, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120240/s12024047.png" /> (i.e. each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120240/s12024048.png" /> is a cycle with simplices with vertices in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120240/s12024049.png" /> of maximal diameter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120240/s12024050.png" />), with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120240/s12024051.png" />, such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120240/s12024052.png" /> by some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120240/s12024053.png" />-chain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120240/s12024054.png" />:

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A Sitnikov-cycle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120240/s12024056.png" /> specifies the chains <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120240/s12024057.png" />. So, two Sitnikov cycles

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give rise to the same Vietoris cycle but eventually to different Sitnikov homology classes.

There have been attempts to extend Steenrod–Sitnikov homology to more general topological spaces, leading to what is called [[Strong homology|strong homology]]. This was initiated for ordinary homology theories by S. Mardešić and Ju. Lisica. There is an analogous axiomatic characterization of strong homology, where one has to replace the cluster axiom by a continuity axiom on the chain level (the c-continuity of a homology theory).

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  F.W. Bauer,   "Extensions of generalized homology theories"  ''Pacific J. Math.'' , '''128''' :  1  (1987)  pp. 25–61</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  S. Eilenberg,   N. Steenrod,   "Foundations of algebraic topology" , ''Princeton Math. Ser.'' , '''15''' , Princeton Univ. Press  (1952)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  J. Milnor,   "On the Steenrod homology theory" , Berkeley  (unpublished)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  K. Sitnikov,   "Combinatorial topology of non-closed sets I"  ''Mat. Sb.'' , '''84''' :  76  (1954)  pp. 3–54  (In Russian)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  K. Sitnikov,   "Combinatorial topology of non-closed sets I–II"  ''Mat. Sb.'' , '''37''' :  79  (1955)  pp. 355–434  (In Russian)</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  N. Steenrod,   "Regular cycles of compact metric spaces"  ''Amer. J. Math.'' , '''41'''  (1940)  pp. 833–85</TD></TR></table>