UDC 513.836

MATHEMATICS

V. R. KIREITOV

FIBERED MODULES AND COBORDISMS

(Presented by Academician A. D. Aleksandrov, 3 XII 1969)

The paper considers questions connected with the specification of a module structure on a vector bundle, and cobordisms based on the specification of module structures in the normal bundle of a manifold. If the field of complex numbers or the skew field of quaternions is taken as the ring of operators of the module structure, then we obtain the notion of a quascomplex or quasisymplectic structure. The corresponding cobordisms are considered in detail in ((^1,{}^4)).

Everywhere below (K) is the field of real or complex numbers, and (\Lambda) is a finite-dimensional associative (K)-algebra with unit. A fibered right (\Lambda)-module is a (K)-vector bundle with a prescribed right action of the algebra (\Lambda) on it by means of (K)-automorphisms of the vector bundle that are identical on the base. A mapping of fibered (\Lambda)-modules is a mapping of vector bundles that commutes with the action of the operators. Fibered (\Lambda)-modules and their mappings form a category.

A fibered (\Lambda)-module (\xi_\Lambda) is called linearly homogeneous if the fibers lying over points of one component of linear connectedness of the base (B_\xi) of the module (\xi_\Lambda) are (\Lambda)-isomorphic; the (\Lambda)-module (A), to which all fibers of the fibered (\Lambda)-module (\xi_\Lambda) lying over some component of linear connectedness are isomorphic, is called a typical fiber over the given component of linear connectedness.

A fibered (\Lambda)-module is called (\Lambda)-trivial if there exists a (\Lambda)-module (A) such that the fibered (\Lambda)-modules (\xi_\Lambda) and (i(A) = (B_\xi \times A, \pi, B_\xi)) are (\Lambda)-isomorphic, where (B_\xi) is the base for (\xi_\Lambda) and (\pi(b,a)=b), (b \in B_\xi), (a \in A). (\xi_\Lambda) is called locally (\Lambda)-trivial if there exists an open covering of its base such that its restriction to any element of the covering is a (\Lambda)-trivial fibered module.

Theorem 1. A fibered (\Lambda)-module (\xi_\Lambda), whose base is a connected (CW)-complex, is locally (\Lambda)-trivial if and only if it is linearly homogeneous.

For certain classes of algebras the question of linear homogeneity of fibered modules over them is easily resolved. For example, if (\Lambda) is a semisimple algebra or (\Lambda = K[X]/{f(X)}), where (K[X]) is the algebra of polynomials in the variable (X) over (K), and ({f(X)}) is the ideal generated by the polynomial (f(X)), then every fibered module is linearly homogeneous. Since the indicated example of algebras is the leading one for us, in the subsequent considerations we restrict ourselves to locally (\Lambda)-trivial fibered modules, which, as follows from the assertions formulated above, is not a restriction for the indicated types of algebras. Thus, unless otherwise stated, by a fibered module we shall mean a locally (\Lambda)-trivial fibered module with base a (CW)-complex.

In the category of fibered (\Lambda)-modules the notions of sum, direct product of fibered modules, and also the notion of the inverse image (f!(\xi_\Lambda)) of a fibered (\Lambda)-module (\xi_\Lambda) under a continuous mapping (f : X \to B_\xi) of a (CW)-complex (X) into the base (B_\xi) of the fibered module (\xi_\Lambda), are defined in the obvious way. As in the case of vector bundles, the following assertion holds: if

(f, g: X \to B_\xi), and (f) is homotopic to (g), then the fibered (\Lambda)-modules (f!(\zeta_\Lambda)) and (g!(\zeta_\Lambda)) are (\Lambda)-isomorphic.

Denote by (R(\Lambda,A)) the category of fibered (\Lambda)-modules with typical fiber (A). For this category there exists a universal fibered (\Lambda)-module. Namely:

Theorem 2. There exists such a fibered (\Lambda)-module (\vartheta(A)) that for every (\zeta_\Lambda \in R(\Lambda,A)) there exists, uniquely up to homotopy, a map (f_\xi: B_\xi \to B_{\vartheta(A)}) such that (\zeta_\Lambda = f!\vartheta(A)).

The proof of the last two assertions is based on the fact that assigning a module structure on the bundle (\zeta) is equivalent to specifying a reduction of the structure group to a certain subgroup. Taking this into account, both assertions follow from analogous assertions for principal fiber spaces with arbitrary structure group.

Let, for example, (K=R), and let (f(X)) be a real polynomial of degree (n) without multiple roots, of which (k) roots are real and (2l) roots are complex ((k+2l=n)); let (A) be the sum of (m) copies of free monogenic modules over the algebra (\Lambda=R[X]/{f(X)}).

Lemma 1. The universal fibered (\Lambda)-module (\vartheta(A)) for the category (R(\Lambda,A)) is (K)-isomorphic to the bundle

[
\left(\prod_{1}^{k}\vartheta_m(R)\right)\times\left(\prod_{1}^{l}\vartheta_m(C)\right),
]

where (\vartheta_m(K)) is the universal (m)-dimensional (K)-vector bundle, and (\Pi,\times) denote the operation of taking the direct sum of bundles.

The case in which the polynomial (f(X)) has multiple roots differs essentially from the preceding case. For example, if (f(X)=X^2), then we have

Lemma 2. The universal fibered (\Lambda)-module (\vartheta(A)), where (A) is the sum of (m) copies of free monogenic (\Lambda)-modules over the algebra (\Lambda=R[X]/{X^2}), is (K)-isomorphic to the vector bundle (\vartheta_m(R)\oplus\vartheta_m(R)), where (\oplus) is the operation of taking the direct sum of bundles.

Now let (\Lambda=K(n)) be the algebra of (n\times n)-matrices, (B) the canonical irreducible (\Lambda)-module (i.e. (B=K^n), the arithmetic space on which (K(n)) acts on the right in the canonical way—row by column), and let (A) be the direct sum of (m) copies of the module (B).

Lemma 3. The universal fibered module for the category (R(\Lambda,A)) is isomorphic to the vector bundle

[
\sum_{1}\vartheta_m(K),
]

where (\sum) denotes here the operation of taking the direct sum of bundles.

Let (A) be a right (\Lambda)-module; (A^m) the sum of (m) copies of the module (A). The series (A,A^2,\ldots,A^m,\ldots) of right (\Lambda)-modules gives rise to the series (R(\Lambda,A)), (R(\Lambda,A^2),\ldots,R(\Lambda,A^m),\ldots) of categories and to the corresponding series of universal fibered (\Lambda)-modules (\vartheta(A),\vartheta(A^2),\ldots,\vartheta(A^m),\ldots). The embeddings of modules (i_m:A^m\to A^{m+1}) induce maps (J_m:\vartheta(A^m)\oplus i(A)\to\vartheta(A^{m+1})) of fibered modules, where (i(A)) is a (\Lambda)-trivial fibered module with fiber (A).

We shall say that a (K)-vector bundle (\xi) admits a (\Lambda)-module structure with typical fiber an isotopic module of type (A), or, more briefly, a stable (A)-structure, if it admits a (\Lambda)-module structure after adding a trivial bundle of some dimension and the obtained fibered (\Lambda)-module belongs to one of the categories (R(\Lambda,A^m)) for some (m). A stable (A)-structure on a bundle (\xi) is the homotopy class of an action of the algebra (\Lambda) on the stable vector bundle associated with (\xi).

Analogously to (1–3), one may define the notion of cobordisms of smooth manifolds with a fixed stable (A)-structure in the stable normal bundle of the manifold under some smooth embedding of it into a sphere of high dimension.

There arise groups (\Omega_i(A)) of (i)-dimensional cobordisms of smooth manifolds with stable (A)-structure. Since the direct product of manifolds equipped with a stable (A)-structure is canonically endowed with a stable (A)-structure, and this is compatible with the cobordism relation,

dism, then the graded group (\Omega_*(A)=\sum_{i\geq 0}\Omega_i(A)) is endowed with the structure of a graded commutative ring with identity.

Lemma 4.
[
\Omega_i(A)=\lim_{m\to\infty}\operatorname{ind}{\pi_{km+i}(T\vartheta(A^m))}
=\pi_i(T\vartheta(A)),\qquad i\geq 0,
]
(m\geq 1,\ k=\dim_R A,\ T\vartheta(A)={T\vartheta(A^m)}) is a superspectrum of Thom complexes of universal stratified modules.

Moreover, since the spectrum (T\vartheta(A)) is multiplicative, the total homotopy group of this spectrum is endowed with the structure of a graded ring and, as follows from the properties of the pairing of the spectrum (T\vartheta(A)), this graded ring is associative, commutative, and has an identity. Let
[
\pi(T\vartheta(A))=\sum_{i\geq 0}\pi_i(T\vartheta(A))
]
be the homotopy ring of the spectrum (T\vartheta(A)). Then the preceding assertion about the groups (\Omega_i(A)) admits a strengthening: the rings (\Omega_(A)) and (\pi(T\vartheta(A))) are isomorphic. Taking for (A) a free monogenic (\Lambda)-module over the algebra (\Lambda=R[X]/{f(X)}), we obtain:
[
\Omega_(A)=\pi\left(\left{T\left(\prod_1^k \vartheta_m(R)\right)\wedge
T\left(\prod_1^l \vartheta_m(C)\right)\right}\right)=
]
[
=\pi\left(\left{\left(\bigwedge_1^k T\vartheta_m(R)\right)\wedge
\left(\bigwedge_1^l T\vartheta_m(C)\right)\right}\right),
]
where (\bigwedge) is the operation of joining spectra, whose definition is given below.

If (A) is an irreducible (K(n))-module, then we obtain
[
\Omega_*(A)=\pi\left(\left{T\left(\sum_1^n \vartheta_m(K)\right)\right}\right).
]

Below we give computations of the homotopy groups of the spectra obtained in the preceding examples. If (X=(X_n,f_n)), (Y=(Y_n,g_n)) are superspectra, then their join is the spectrum
[
X\bigwedge Y=(X_n\bigwedge Y_n,\,f_n\bigwedge g_n).
]
The group of (X)-homology of the spectrum (Y) is the group
[
X(Y)=\sum_{i\geq 0}X_i(Y),\qquad
X_i(Y)=\lim_{n\to\infty}\operatorname{ind}({X_{n+i}(Y_n^{\cdot})}).
]
Here the homology theory generated by a spectrum is denoted by the same symbol as the spectrum. If the spectra (X) and (Y) are both multiplicative, then the spectrum (X\bigwedge Y) is also multiplicative, and therefore the group (\pi(X\bigwedge Y)) is a graded ring. In this case the group (X(Y)) is also a graded ring.

Theorem 3. The rings (\pi(X\bigwedge Y)) and (X(Y)) are isomorphic.

This theorem makes it possible to compute completely the homotopy rings
[
\pi\left(\left(\bigwedge_1^k T\vartheta(R)\right)\wedge
\left(\bigwedge_1^l T\vartheta(C)\right)\right),
]
and hence the corresponding cobordism rings.

Let, for example, (A) be a free monogenic module over the algebra
[
\Lambda=R[X]/{f(X)},\qquad f(X)=X^4+1.
]
Then
[
\Omega_*(A)=\Omega_U[x_1,x_2,\ldots,x_n,\ldots]
]
is the polynomial ring in the variables (x_1,x_2,\ldots,x_n,\ldots), (\dim x_n=2n), over the ring (\Omega_U) of (U)-cobordisms of a point.

If (A) is taken as before, (\Lambda=R[X]/{f(X)}), (f(X)=X^3+1), then the ring
[
\Omega_*(A)=\Omega_O[x_1,x_2,\ldots,x_n,\ldots],
]
where (\Omega_O) is the ring of (O)-cobordisms of a point.

One can also indicate geometric generators for cobordism rings of this kind. The study of spectra of the form
[
\left{T\left(\sum_1^n \vartheta_m(K)\right)\right}
]
leads to the following (restricting ourselves to the case (K=C)):
[
\operatorname{rk}\left(\pi_{2i}\left(T\left(\sum_1^n\vartheta_m(C)\right)\right)\otimes Q\right)
=
\operatorname{rk}\left(\Omega_{2i}^U\otimes Q\right),
]

[
\operatorname{rk}\left(\pi_{2i+1}\left(T\left(\sum_{1}^{n}\vartheta_m(C)\right)\otimes Q\right)\right)=0,
]

where (Q) is the field of rational numbers.

The computation of the twisted parts of these homotopy groups, however, encounters difficulties. Namely, let, for example, (X={T(\vartheta_m(C)\oplus \vartheta_m(C))}). The cohomology with coefficients in the group (Z_2) of the space (T(\vartheta_m\otimes\vartheta_m)) is mapped isomorphically onto the ideal of the ring (H^*(BU_m;Z_2)) generated by the element (C_m^2), where (C_m) is the top Chern class of this ring. We have the relation (Sq^4 C_m^2=C_1^2 C_m^2), (Sq^2(C_1 C_m^2)=C_1^2 C_m^2=Sq^4 C_m^2), and at the same time (C_1 C_m^2) is not the value of any cohomology operation on the element (C_m^2). This means that the cohomology of the spectrum (X), regarded as a module over the Steenrod algebra, is not representable as a sum of monogenic modules, as happens in the classical case.

Novosibirsk
State University

Received
20 XI 1969

REFERENCES

1. S. P. Novikov, Matem. sborn., 57, 99, no. 4 (1962).
2. R. Lashof, Trans. Am. Math. Soc., 109, no. 2 (1963).
3. J. Milnor, L’Enseignement Mathematic, 8, 16 (1962).
4. J. Milnor, Am. J. Math., 82, no. 3, 505 (1960).