Fredholm Complexes on
by
Sadi Bayramov
Kocaeli University, Izmit

The object of this study is to provide an alternative description of the K-functor introduced in [5], by utilizing the ideas presented in [1] and [4]. Note that this problem was studied initially in works [2] and [3].

Let P(\Lambda) be a category of finite-induced projective \Lambda-modules and let Vec(X, \Lambda) be a category of local trival vector bundles over a compact space X, such that the bundle's fibers belong in the category P(\Lambda). If a compact Lie Group G acts on the space X then the category of G-\Lambda-bundles will be denoted by Vec t_G(X, \Lambda). Finally, let Vec t_{G, \infty}(X, \Lambda) be a category of local trivial bundles whose fibers do not necessarily belong in the category P(\Lambda).

Now, let us consider the following cochaineed complex of bundles over a compact space X, that are members of the category Vec t_{G, \infty}(X, \Lambda):

(E, d)=0 --> E(0) d0 -->   E(1) d1 -->   ... dn-1 -->   E(n) --> 0      (1)

Definition 1. Cochaned complex (1) is called a Fredholm complex of bundles if there exist homomorphisms h_i : Eⁱ --> E^i-1 satisfying the condition

hi+1 di + di-1 hi = 1-Ki.                                                (2)

where K is compact \Lambda-operators.

Fred_G (X, \Lambda) denotes a category of Fredholm complexes of bundles over X and CC Vect_G(X, \Lambda) denotes a category of cochained complexes of G-\Lambda- bundles.

Definition 2. i) Cochained complexes of bundles (E, d) and (E', d') over the space X are called chain-equivalent if there exist morphisms f: (E, d) --> (E', d'), g:(E', d') --> (E, d) satisfying the conditions:

f o g \approx 1(E, d),     g o f \approx 1E', d'f).

Chain-equivalence is denoted as follows: (E, d) \approx (E', d').

ii) Cochained complexes (E, d) and (E', d') are said to be homotopic (to each other) if there exists a complex (F, d^'') over the space X ×[0, 1] that satisfies the following conditions:

(F, d'') |X ×0 = (E, d),   (F, d'') |X ×1 = (E', d').

Homotopy is denoted as follows: (E, d) =~ (E', d').

iii Complexes (E, d) and (E', d') are said to be equivalent (to each other) if there exist acyclic complexes (F, d) and (F', d') such that

(E, d) \oplus(F, d) =~ (E', d') \oplus(F', d').

Equivalence is denoted as follows: (E, d) ~ (E', d').

Theorem 1. Categories Fred_G(X, \Lambda) and CC Vect_G(X, \Lambda) are chained homotopy equivalent.

Theorem 2. Let (F, d) denote a Fredholm complex of bundles over a compact space X. There exist (E, d) in CC Vect_G(X, \Lambda) and acyclic complexes (A₀, d₀) and A₁, d₁) such that

(E, d) \oplus(A0, d0) ~ (F, d) \oplus(A1, d1).

Corollary 1. The group K_G (X, \Lambda) is the Grothendieck completion of the semi-group of Fredholm complexes of bundles.

References

1. M. F. Atiyah, K-theory, Benjamin, 1967.
2. A. S. Bayramov, Index Theory for Fredholm complexes on C^*-algebras, Proceedings of the International Topology Conference, Baku, 1986.
3. M. R. Bunyatov and S. A. Bayramov, On Index Theory for a family of Fredholm complexes, Amer. Math. Soc. Transl., 154, 2, 1992, 171-177.
4. G. Segal, Fredholm Complexes, Quart. J. Math. Oxford 2, 21, 1970, 385-402.
5. E. V. Troitsky, The Rickvivaryant index of C^*- elliptic operators, Izv. Akad. Nauk. SSSR, 50, 4, 1986, 849-865.

Date received: July 23, 2001