On a Vietoris-Begel Theorem for non-compact spaces

James M. Parks

Abstract The Vietoris-Begel Theorem of A. Bialynicki-Birula [4] is extended to non-compact spaces X by requiring additional acyclicity of fibers on the remainder NX = BX-X of the Stone-Cech compactification BX of X. This extends results of Bialynicki-Birula [4] and Skliarenko [2] to spaces X which need not be normal.

All spaces will be assumed to be completely regular and Hausdorff.

The notation BX and Bf will denote the Stone-Cech compactification of X and the Stone extension of the map f:X->Y to the map Bf:BX->BY, respectively. Let p denote the natural embedding of a space X, into the Stone-Cech compactification of X, BX, p:X->BX. Also, NX = BX-X will denote the remainder of X in BX.

Although it is possible to use sheaf cohomology to obtain the results below (see [4]), this will not be used here. We prefer to use Cech (Alexander) cohomology with coefficients in an abelian group A, denote H*(X, A) [1].

The main result is the following theorem (cf. [4]).

Theorem Let f:X -> Y and g:Y -> Z be onto maps, all spaces completely

regular and Hausdorf, such that for all x in Z the homomorphism

f*:H^k(Bg^-1(x), A) -> H^k(Bf^-1Bg^-1(x), A)

is an isomorphism for k = 0, . . . , n and a monomorphism for k = n+1.

Suppose also that H^k(Bf^-1(y), A) = 0 for all y in Bg^-1(x), x in NZ, and

k = 0, . . . , n.

Then f*:H^k(Y, A) -> H^k(X, A) is an isomorphism for

k = 0, . . . , n and a monomorphism for k = n+1.

Proof:

First note that since the composition pf:X -> BY is a map into a compact space,

it has the Stone extension Bpf: BX -> BY. Thus Bpf(BX) is a compact set in BY

which contains the dense set Y, hence Bpf must be onto ([3], Th.6.7).

If x in Z, then Bg^-1(x) = Cl(g^-1(x) in BX and thus

Bf*:H^k(Bg^-1(x), A) -> H^k(Bf^-1Bg^-1(x), A)

is an isomorphism for k = 0, . . . , n and a monomorphism for k = n+1 by the

assumption on f* and the properties of Cech theory [1].

If x in NZ, then

Bf*:H^k(Bg^-1(x), A) -> H^k(Bf^-1Bg^-1(x), A)

is an isomorphism for k = 0, . . . , n and a monomorphism for k = n+1 by the

Vietoris-Begel Theorem ([5], Th.11.1).

Hence if x in Z,

Bf*:H^k(Bg^-1(x), A) -> H^k(Bf^-1Bg^-1(x), A)

is an isomorphism for k = 0, . . . , n and a monomorphism for k = n+1.

Thus the hypothesis of Bialynicki-Birula's Theorem ([4], Th.1) is satisfied and

Bf*:H^k(BY, A) -> H^k(BX, A)

is an isomorphism for k = 0, . . . , n and a monomorphism for k = n+1.

The result then follows from the well known invariance of Cech cohomology on the Stone-Cech compactification of completely regular Hausdorff spaces ([1], p.282).

The Vietoris-Begel Theorem [4] follows immediately as a corollary by letting Y = Z and g = 1_Y.

A simple application to Cech continuity is obtained as follows.

Let {X_n, a_mⁿ} be an inverse system of completely regular, Hausdorff spaces with inverse limit space X. Let {BX_n, Ba_mⁿ} be the associated inverse system with limit space X'.

Then X' contains a copy of X, say X".

Let NX" = X' - X" denote the remainder of X" in X'.

Let j:X -> X' be the embedding determined by the embeddings j_n: X_n -> BX_n

([6] p.430).

Then we have the following corollary.

Corollary If X, X' and NX" are as above and H^k(Bj^-1(x), A) = 0, for all k = 1, . . . ,

and all x in NX", where Bj: BX -> X', then H^k(X, A) is isomorphic to L_-->nH^k(X_n,A), for all k.

Proof:

The quotient map Bj: BX->X' is the Stone extension of j to BX and is onto, closed, and proper ([6] p.226).

By the theorem above Bj*:H^k(X', A) -> H^k(BX, A) is an isomorphism for all k.

However, H^k(X', A), L_-->nH^k(BX_n, A), and L_-->nH^k(X_n, A) are all isomorphic by Cech

continuity and Cech invariance ([1] p.259), hence the result follows.

Bibliography

1. Eilenberg, S., and N. Steenrod, Foundations of Algebraic Topology, Princeton (1952).

2. Skliarenko, E. G., Continuation of Homeomorphisms, Dokl. Akad. Nauk SSSR, 141(1961), pp. 1045-1047.

3. Gillman, L., and M. Jerison, Rings of Continuous Functions, Van Nostrand (1960).

4. Bialynicki-Birula, A., On Vietoris Mapping Theorem and Its Inverse, Fund. Math. 53(1964), pp.135-145.

5. Bredon, G. E., Sheaf Theory, McGraw-Hill (1967).

6. Dugundji, J., Topology, Allyn and Bacon (1967).

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