A Generalization of Seifert-Van Kampen Theorem for Fundamental Groups

Let X be a topological space. A fundamental group π 1 (X, x 0 ) of X based at a point x 0 ∈ X is formed by homotopy arc classes in X based at x 0 ∈ X. For an arcwise-connected space X, it is known that π 1 (X, x 0 ) is independent on the base point x 0 , that is, for ∀x 0 , y 0 ∈ X, π 1 (X, x 0 ) ∼ = π 1 (X, y 0 ).

Find the fundamental group of a space X is a difficult task in general. Until today, the basic tool is still the Seifert-Van Kampen theorem following.

Theorem 1.1(Seifert and Van-Kampen) Let X = U ∪ V with U, V open subsets and let X, U, V , U ∩ V be non-empty arcwise-connected with x 0 ∈ U ∩ V and H a group. If there are homomorphisms φ 1 : π 1 (U, x 0 ) → H and φ 2 : π 1 (V, x 0 ) → H and with φ 1 • i 1 = φ 2 • i 2 , where i 1 : π 1 (U ∩ V, x 0 ) → π 1 (U, x 0 ), i 2 : π 1 (U ∩ V, x 0 ) → π 1 (V, x 0 ), j 1 : π 1 (U, x 0 ) → π 1 (X, x 0 ) and j 2 : π 1 (V, x 0 ) → π 1 (X, x 0 ) are homomorphisms induced by inclusion mappings, then there exists a unique homomorphism Φ : π 1 (X, x 0 ) → H such that Φ • j 1 = φ 1 and Φ • j 2 = φ 2 .

Applying Theorem 1.1, it is easily to determine the fundamental group of such spaces X = U ∪ V with U ∩ V an arcwise-connected following.

Theorem 1.2(Seifert and Van-Kampen theorem, classical version) Let spaces X, U, V and x 0 be in Theorem 1.1. If j : π 1 (U, x 0 ) * π 1 (V, x 0 ) → π 1 (X, x 0 ) is an extension homomorphism of j 1 and j 2 , then j is an epimorphism with kernel Kerj generated by i -1 1 (g)i 2 (g), g ∈ π 1 (U ∩ V, x 0 ), i.e.,

, where [A] denotes the minimal normal subgroup of a group G included A ⊂ G . Now we use the following convention.

denote homomorphisms induced by the inclusion mapping U λ → U µ and U λ → X, respectively. It should be noted that the Seifert-Van Kampen theorem has been generalized in [2] under Convention 1.3 by any number of open subsets instead of just by two open subsets following.

Theorem 1.4([2]) Let X, U λ , λ ∈ Λ be arcwise-connected space with Convention 1.3 satisfies the following universal mappping condition: Let H be a group and let ρ λ : π 1 (U λ , x 0 ) → H be any collection of homomorphisms defined for all λ ∈ Λ such that the following diagram is commutative for U λ ⊂ U µ : Ţhen there exists a unique homomorphism Φ : π 1 (X, x 0 ) → H such that for any λ ∈ Λ the following diagram is commutative: Moreover, this universal mapping condition characterizes π 1 (X, x 0 ) up to a unique isomorphism.

Theorem 1.4 is useful for determining the fundamental groups of CW-complexes, particularly, the adjunction of n-dimensional cells to a space for n ≥ 2. Notice that the essence in Theorems 1.2 and 1.4 is that ∩ λ∈Λ U λ is arcwise-connected, which limits the application of such kind of results. The main object of this paper is to generalize the Seifert-Van Kampen theorem to such an intersection maybe nonarcwise connected, i.e., there are

This enables one to determine the fundamental group of topological spaces, particularly, combinatorial manifolds introduced in [6]- [8] following which is a special case of Smarandache multi-space ([9]- [10]).

which enables us to generalize the conception of manifold to combinatorial manifold, also a locally combinatorial Euclidean space following.

Definition 1.5 For a given integer sequence

) is finite if it is just combined by finite manifolds without one manifold contained in the union of others.

A topological graph G is itself a topological space formally defined as follows.

Definition 2.1 A topological graph G is a pair (S, S 0 ) of a Hausdorff space S with its a subset S 0 such that (1) S 0 is discrete, closed subspaces of S;

(2) S -S 0 is a disjoint union of open subsets e 1 , e 2 , • • • , e m , each of which is homeomorphic to an open interval (0, 1);

(3) the boundary e ie i of e i consists of one or two points. If e ie i consists of two points, then (e i , e i ) is homeomorphic to the pair ([0, 1], (0, 1)); if e ie i consists of one point, then (e i , e i ) is homeomorphic to the pair (S 1 , S 1 -{1});

(4) Ņotice that a topological graph maybe with semi-edges, i.e., those edges e + ∈ E(G) with e + : [0, 1) or (0, 1] → S. A topological space X attached with a graph G is such a space X ⊙ G such that

and there are semi-edges e + ∈ (X G) \ G, e + ∈ G \ X. An example for X ⊙ G can be found in Fig. 2.1. In this section, we characterize the fundamental groups of such topological spaces attached with graphs.

Theorem 2.2 Let X be arc-connected space, G a graph and

Proof This result is an immediately conclusion of Seifert-Van Kampen theorem. Let U = X and V = G. Then X ⊙ G = X ∪ G and X ∩ G = H. By definition, there are both semi-edges in G and H. Whence, they are

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