D. O. BALADZE

GENERALIZED STEENROD HOMOLOGY GROUPS

(Presented by Academician P. S. Aleksandrov on 29 XI 1968)

Let \(L\) and \(L'\) be some simplicial complexes. A mapping \(f: L \to L'\) will be called a closed simplicial embedding if it is a simplicial mapping that maps \(L\) homeomorphically onto some closed subcomplex of the complex \(L'\).

We shall consider a certain category \(\mathfrak A\), whose objects are locally finite simplicial complexes of a specified kind, and whose mappings are closed simplicial embeddings of a specified kind. In particular, for every complex \(L\) that is an object of the category \(\mathfrak A\), the mapping \(1_L\), which is the identity mapping of the complex \(L\) onto itself, is defined in this category. In what follows we shall assume that the category \(\mathfrak A\) has the following properties:

\(1^\circ\). If \(L_1\) and \(L_2\) are arbitrary objects of the category \(\mathfrak A\), then the union \(L_1 \cup L_2\) (without identifications) is also an object of the category \(\mathfrak A\), and the natural mappings \(L_1 \to L_1 \cup L_2\) and \(L_2 \to L_1 \cup L_2\) belong to the category \(\mathfrak A\). More precisely, this means the following: for any objects \(L_1, L_2 \in \mathfrak A\) there are in the category \(\mathfrak A\) an object \(L\) and mappings \(f_1: L_1 \to L\), \(f_2: L_2 \to L\) such that \(f_1(L_1) \cap f_2(L_2) = \varnothing\) and \(f_1(L_1) \cup f_2(L_2) = L\) (recall that every mapping of the category \(\mathfrak A\) is a closed simplicial embedding and, in particular, a homeomorphism).

\(2^\circ\). If \(f: L \to L_1\) and \(g: L \to L_2\) are mappings belonging to the category \(\mathfrak A\), then the complex \(L^*\), composed of \(L_1\) and \(L_2\), having common part \(f(L) = g(L)\), also belongs to the category \(\mathfrak A\), and the natural mappings \(L_1 \to L^*\), \(L_1 \to L^*\) belong to the category \(\mathfrak A\). More precisely, this means the following:

If \(f: L \to L_1\) and \(g: L \to L_2\) are mappings belonging to the category \(\mathfrak A\), then there exists in the category \(\mathfrak A\) a complex \(L^*\) and mappings \(\alpha: L \to L^*\), \(\beta: L_1 \to L^*\), \(\gamma: L_2 \to L^*\), for which \(\beta \circ f = \gamma \circ g = \alpha\), \(\beta(L_1) \cup \gamma(L_2) = L^*\), and, moreover, for zero-dimensional skeleta the equality \(\beta(L_1^0) \cap \gamma(L_2^0) = \alpha(L^0)\) holds. In other words, the complex \(L^*\) is the union of its two subcomplexes \(\beta(L_1)\) and \(\gamma(L_2)\), the intersection of which contains the subcomplex \(\alpha(L)\) (although it may not coincide with it), while the subcomplexes \(\beta(L_1)\) and \(\gamma(L_2)\) have no common vertices except those that belong to the subcomplex \(\alpha(L)\).

Suppose further that some group \(G\) is fixed and that, in each complex \(L\) which is an object of the category \(\mathfrak A\), a certain class of chains \(\alpha(L)\) over the coefficient group \(G\) is specified and a differential operator \(d\) is defined, satisfying the condition \(d \circ d = 0\), and that the following conditions are fulfilled (the lower indices indicate a certain filtration, hereafter called dimension):

1. If \(x_p \in \alpha(L)\), then also \(d x_p \in \alpha(L)\).
2. If \(x_p, y_p \in \alpha(L)\), then \(x_p + y_p \in \alpha(L)\).
3. If \(x_p \in \alpha(L)\) and \(g\) is some endomorphism of the group \(G\), then \(g x_p \in \alpha(L)\).
4. If \(x_p \in \alpha(L)\) and \(f: L \to L_1\) is a mapping of the category \(\mathfrak A\), then \(f(x_p) \in \alpha(L_1)\).

We note that if \(K\) is some ring and \(G\) is the additive group of this ring, then the correspondence \(g\mapsto kg\) for any \(k\in K\) defines an endomorphism of the group \(G\). Thus, in this case it follows from axiom 3, in particular, that if \(x_p\in a(L)\), then also \(kx_p\in a(L)\) for any \(k\in K\). Moreover, from axiom 3 it follows (in the case of any abelian group \(G\)) that if \(x_p\in a(L)\), then \(-x_p\in a(L)\). Together with axiom 2 this means that all \(p\)-dimensional chains belonging to \(a(L)\) form an abelian group \(a_p(L)\) (with respect to the usual addition of chains). Thus the mappings
\(d:a_p(L)\to a_{p-1}(L)\) and \(f:a_p(L)\to a_p(L_1)\) (for any mapping \(f:L\to L_1\) belonging to the category \(\mathfrak A\)) are homomorphisms.

Finally, let \(L\) be an arbitrary object of the category \(\mathfrak A\), and let \(\varphi\) be some mapping of its zero-dimensional skeleton \(L^0\) into a compact metric space \(\Phi\). We shall call the mapping \(\varphi\) regular (cf. \((^1)\)) if, for every \(\varepsilon>0\), only a finite number of simplices of the complex \(L\) have their vertices mapped into a subset of the compactum \(\Phi\) whose diameter is less than \(\varepsilon\).

If \(L\in\mathfrak A\), \(x_p\in a_p(L)\), and \(\varphi:L^0\to\Phi\) is a regular mapping, then the triple \((L,\varphi,x_p)\) will be called a \(p\)-dimensional regular \((\mathfrak A,a)\)-chain of the space \(\Phi\) over the coefficient group \(G\). If \(dx_p=0\), then the triple \((L,\varphi,x_p)\) will be called a regular \(p\)-dimensional \((\mathfrak A,a)\)-cycle of the space \(\Phi\) over the group \(G\).

Two \(p\)-dimensional regular \((\mathfrak A,a)\)-cycles \((L_1,\varphi_1,x_p')\) and \((L_2,\varphi_2,x_p'')\) of the space \(\Phi\) over the group \(G\) are called homologous to each other if there exist mappings \(f_1:L_1\to L\), \(f_2:L_2\to L\) of the category \(\mathfrak A\) and a \((p+1)\)-dimensional regular \((\mathfrak A,a)\)-chain \((L,\varphi,x_{p+1})\) such that \(\varphi\) coincides with \(\varphi_1\) on \(f_1(L_1)\) (i.e. \(\varphi\circ f_1=\varphi_1\)) and with \(\varphi_2\) on \(f_2(L_2)\) (i.e. \(\varphi\circ f_2=\varphi_2\)), and
\(dx_{p+1}=f_1(x_p')-f_2(x_p'')\).

It follows from this definition in a trivial way that if \((L,\varphi,x_p)\) is some \(p\)-dimensional regular \((\mathfrak A,a)\)-cycle of the space \(\Phi\) over the group \(G\), and \(f:L\to L_1\) is a mapping of the category \(\mathfrak A\), and there exists a regular mapping \(\varphi_1:L_1^0\to\Phi\) satisfying the condition
\(\varphi_1\circ(f|L^0)=\varphi\), then the \(p\)-dimensional regular cycles \((L,\varphi,x_p)\) and \((L_1,\varphi_1,f(x_p))\) are homologous to each other.

Next, it is easy to show that the homology relation is reflexive, symmetric, and transitive (cf. \((^1)\)). Thus the \(p\)-dimensional regular \((\mathfrak A,a)\)-cycles \((L,\varphi,x_p)\), \(x_p\in a(L)\), of the space \(\Phi\) over the group \(G\) split into disjoint classes of mutually homologous regular \((\mathfrak A,a)\)-cycles. These classes are called homology classes of regular \(p\)-dimensional \((\mathfrak A,a)\)-cycles of the space \(\Phi\) over \(G\).

The sum of two \(p\)-dimensional regular \((\mathfrak A,a)\)-chains \((L_1,\varphi_1,x_p')\) and \((L_2,\varphi_2,x_p'')\) is called the \(p\)-dimensional regular \((\mathfrak A,a)\)-chain \((L_1\cup L_2,\varphi,x_p'+x_p'')\), where \(L_1\cup L_2\) is the union of the complexes \(L_1\) and \(L_2\) without identifications, and the mapping \(\varphi\) coincides with \(\varphi_1\) on \(L_1\) and with \(\varphi_2\) on \(L_2\). More precisely, this definition can be formulated as follows. Let \((L_1,\varphi_1,x_p')\) and \((L_2,\varphi_2,x_p'')\) be two \(p\)-dimensional regular \((\mathfrak A,a)\)-chains. Let, further, \(L\) be an object of the category \(\mathfrak A\), and let \(f_1:L_1\to L\), \(f_2:L_2\to L\) be mappings of the category \(\mathfrak A\) satisfying the conditions
\(f_1(L_1)\cup f_2(L_2)=L\),
\(f_1(L_1)\cap f_2(L_2)=\varnothing\). Denote by \(\varphi:L\to\Phi\) the mapping satisfying the relations
\(\varphi\circ f_1=\varphi_1\), \(\varphi\circ f_2=\varphi_2\) (under these conditions the mapping \(\varphi\), evidently, is defined and moreover uniquely). Next, put
\(x_p=f_1(x_p')+f_2(x_p'')\). We then obtain the \(p\)-dimensional regular \((\mathfrak A,a)\)-chain
\[ (L,\varphi,x_p)=\bigl(L,\varphi,f_1(x_p')+f_2(x_p'')\bigr), \]
which is called the sum of the two \((\mathfrak A,a)\)-chains under consideration. (By this definition the sum is specified uniquely up to equivalence.)

The sum of two homology classes of regular \(p\)-dimensional \((\mathfrak A,a)\)-cycles is defined uniquely as the homology class containing the sum of two \(p\)-dimensional regular \((\mathfrak A,a)\)-cycles respectively belonging to the summands.

classes under consideration. The addition of homology classes thus defined turns the totality of all homology classes of \(p\)-dimensional regular \((\mathfrak A,a)\)-cycles into a commutative group, which we shall denote by \(H_p^{(\mathfrak A,a)}(\Phi,G)\).

The group \(H_p^{(\mathfrak A,a)}(\Phi,G)\) will be called, by definition, the \(p\)-dimensional generalized homological Steenrod group of the space \(\Phi\) over the coefficient group \(G\), or, otherwise, the \(p\)-dimensional \((\mathfrak A,a)\)-homology group of the space \(\Phi\) over the coefficient group \(G\).

It is easy to see that if for \(\mathfrak A\) one takes the category of all locally finite complexes and all their closed simplicial embeddings, and for \(a(L)\) one takes the totality of all (infinite) chains of the complex \(L\) over the coefficient group \(G\) (with the usual understanding of dimension), then in this case the \((\mathfrak A,a)\)-homology group coincides with the Steenrod group defined in \((^{1})\). In other words, the Steenrod groups can be obtained as a special case of the construction described above. It is precisely this fact that is the reason why the \((\mathfrak A,a)\)-homology groups defined above have been called generalized Steenrod homological groups.

Moreover, as we shall show in the following note, many of the previously defined homology groups (see \((^{2-4})\)) are obtained as special cases of the construction described above. For example, as special cases of this construction one can obtain the \(K\)-homology groups defined by the author earlier in \((^{3})\), and hence also Sitnikov’s homological groups \(\Delta_p(\Phi,G)\), defined by him in \((^{4})\). Thus, the homologies considered by K. A. Sitnikov do not fall outside the circle of Steenrod’s ideas.

Received
12 XI 1968

CITED LITERATURE

\(^{1}\) N. E. Steenrod, UMN, 2, issue 2(18), 56 (1947).
\(^{2}\) D. O. Baladze, Reports of the Academy of Sciences of the Georgian SSR, 50, No. 1, 15 (1968).
\(^{3}\) D. O. Baladze, Reports of the Academy of Sciences of the Georgian SSR, 52, No. 2, 283 (1968).
\(^{4}\) K. A. Sitnikov, Mat. sbornik, 34(76), No. 1, 3 (1954).