Fundamental Groups of Homology Theory

INTRODUCTION

Homology, in mathematics, a fundamental belief of algebraic topology. Intuitively, two curves in an aircraft or other -dimensional surface are homologous if together they certain a location—thereby distinguishing between an internal and an outdoor. Similarly, two surfaces inside a three-dimensional area are homologous if together they sure a three-dimensional place lying in the ambient area. There are many ways of creating this intuitive notion specific. The first mathematical steps had been taken within the 19th century by way of the German Bernhard Riemann and the Italian Enrico Betti, with the introduction of ―Betti numbers‖ in each dimension, relating to the number of unbiased (certainly described) gadgets in that size that are not boundaries. Informally, Betti numbers talk to the quantity of instances that an item can be ―reduce‖ earlier than splitting into separate pieces; as an instance, a sphere has Betti quantity 0 on the grounds that any reduce will split it in, even as a cylinder has Betti no 1 considering that a reduce along its longitudinal axis will merely result in a rectangle. A extra huge remedy of homology turned into carried out in n dimensions at the start of the twentieth century by means of the French mathematician Henri Poincaré, main to the perception of a homology group in each measurement, seemingly first formulated approximately 1925 by way of the German mathematician Emmy No ether. The two fundamental information approximately homology organizations for a surface or a better-dimensional topological manifold are: (1) if the businesses are described with the aid of a triangulation, a mobile subdivision, or different artifact, the ensuing businesses do now not depend on the particular alternatives made alongside the way; and (2) the homology companies are a topological invariant, in order that if two surfaces or better-dimensional spaces are homeomorphic, then their homology organizations in every measurement are isomorphic (see foundations of mathematics: Isomorphic structures and arithmetic: Algebraic topology). Homology performs a essential position in analysis; certainly, Riemann become brought about it via questions related to integration on surfaces. The simple cause is because of Green‘s theorem (see George Green) and its generalizations, which express positive integrals over a website in terms of integrals over the boundary. As a consequence, certain vital integrals over curves may have the identical value for any curves which might be homologous. This is in flip pondered in physics in the examine of conservative vector spaces and the existence of potentials.

Stillwell 1993, In mathematics, homology is a popular way of associating a series of algebraic items such as abelian organizations or modules to other mathematical items which include topological

summary algebra, businesses, Lie algebras, Galois Theory, and algebraic geometry. The original motivation for outlining homology agencies become the observation that two shapes may be outstanding by examining their holes. For instance, a circle is not a disk due to the fact the circle has a hollow via it even as the disk is stable, and the ordinary sphere is not a circle due to the fact the sphere encloses a -dimensional hole while the circle encloses a one-dimensional hollow. However, because a hollow is "not there", it is not immediately obvious how to outline a hollow or how to differentiate distinctive forms of holes. Homology became at the beginning a rigorous mathematical method for outlining and categorizing holes in a manifold. Loosely talking, a cycle is a closed sub manifold, a boundary is a cycle which is also the boundary of a sub manifold, and a homology elegance (which represents a hole) is an equivalence elegance of cycles modulo boundaries. A homology class is thus represented by a cycle which isn't the boundary of any sub manifold: the cycle represents a hollow, particularly a hypothetical manifold whose boundary could be that cycle, but which is "not there". There are many exclusive homology theories. A particular kind of mathematical object, consisting of a topological area or a group, can also have one or more associated homology theories. When the underlying item has a geometrical interpretation as topological spaces do, the nth homology group represents behavior in dimension n. Most homology agencies or modules may be formulated as derived elements on suitable abelian classes, measuring the failure of a component to be genuine. From this abstract attitude, homology agencies are decided by way of items of a derived category. Weibel (1999) Homology idea can be stated first of all the Euler polyhedron formula, or Euler function. This was followed by Riemann's definition of genus and n-fold connectedness numerical invariants in 1857 and Betti's proof in 1871 of the independence of "homology numbers" from the choice of basis. Richeson (2008) Homology itself became developed as a way to examine and classify manifolds consistent with their cycles – closed loops (or greater usually sub manifolds) that may be drawn on a given n dimensional manifold but now not continuously deformed into every different. These cycles are also every so often idea of as cuts which may be glued again together, or as zippers which may be mounted and loose. Cycles are classified through measurement. For instance, a line drawn on a surface represents a 1-cycle, a closed loop or (1-manifold), whilst a surface reduces thru a three-dimensional manifold is a 2-cycle. We define here the notion of linear graph and prove the basic theorem that any covering space of a linear graph is itself a linear graph. Recall that an arc A is a space homeomorphic to the unit interval [0, 1. The end points of A are the points p, q corresponding to 0 and 1 under the homeomorphism; they are the unique points of A such that A – p and A – q are connected. The interior of an area A consists of A with its end points deleted.. Definition 1:- A linear graph is a space X that is written as the union of a collection of subspaces A, each of which is an arc, such that: (1) The intersection A  A of two arcs is either empty or consists of a single point that is an end point of each. (2) The topology of X is coherent with the subspaces A. The arcs A are called the edges of X, and their interiors are called the open edges of X. Their end points are called the vertices of X; we denote the set of vertices of X by X0. It X is a linear graph, and if C is a subset of X that equals a union of edges and vertices of X, then C is closed in X. For the intersection of C with A is closed in A, since it is either empty, or it equals A, or it equals one or both vertices of A. It follows that each edge of X is a closed subset of X. It also follows that X0 is a closed discrete subspace of X, since any subset of X0 is closed in X. In the case of finite graph, considered earlier, we used the Hausdorff condition in our definition in place of condition (2); it followed, in that case, that the topology of X was coherent with the subspaces A. In the case of an infinite graph, this would no longer be true, so we must assume the coherence condition as part of the definition. We would assume the Hausdorff condition as well, but it is no longer necessary, for it follows from the coherence condition: Lemma 1:- Every linear graph X is Hausdorff; in fact, it is normal. Proof. Let B and C be disjoint closed subsets of X. Assume, without loss of generality, that every vertex of X belongs either to B or to C. For each , choose disjoint subsets U and V of A that are open in A, containing B  A and C  A, We show the sets U and V are disjoint. If x  U  V, then x  U  V for some   . This fact implies that A and A contain the point x, which means that x is a vertex of X. This is impossible, for if x  B, the x lies in no set V, and if x  C, then x lies in no set

U.

Now we show U and V are open in X. To show U is open, we show that U  A = U for each . By definition, U  A contains U. If x is a point of U  A not in U, then x belongs to U for some   . Then both A and A contain x, so that x must be a vertex of X. This is impossible, for if x  B, then x  U definition of U, and if x  C, then x cannot belong to U. Definition 2 :- Let X be a linear graph. Let Y be a subspace of X that is a union of edges of X. Then Y is closed in X and is itself a linear graph; we call it subgroup of X. To show that Y is a linear graph, we need to show that the subspace topology on Y is coherent with the set of edges of Y. If the subset D of Y is closed in the subspace topology, then D is closed in X, so that D  A is closed in A for each edge of X, and in particular for each edge of Y. Conversely, suppose D  A is closed in A for each edge A of Y. We must show that D  A is closed in A for each edge A of X that is not contained in Y. But in this case, D  A is either empty or a one-point set. We conclude that Y has the topology coherent with its set of edges. Lemma 2:- Let X is a linear graph. If C is a compact subspace of X, there exists a finite sub graph Y of X that contains C. If C is connected, Y can be chosen to be connected. Proof. First, note that C contains only finitely many vertices of X. For C X0 is a closed discrete subspace of the compact space C; since it has no limit point, it must be finite. Similarly, there are only finitely many values of  for which C contains an interior point of the edge A. For if we choose a point x of C interior to A for each index  for which it is possible to do so, we obtain a collection B = {x} whose intersection with each edge A is a one-point set or empty. It follows that every subset of B is closed in X, so that B is a closed discrete subspace of C and hence finite. Form Y by choosing, for each vertex x of X belonging to C, an edge of X having x as a vertex, and adjoining to these edges all edges A whose interiors contain points of C. They Y is a finite sub graph containing C. Note that if C is connected, then Y is the union of a collection of arcs each of which intersects C, so that Y is connected. Proof. Step I. We show X is locally path connected. If x  X and x lies interior to some edge of X, then within every neighborhood of x is a neighborhood of x homeomorphic to an open interval of , which is path connected. On the other hand, if x is a vertex of X and U is a neighborhood of x, then we can choose, for each edge A having x as an end point, a neighborhood V of x in A lying in U that is homeomorphic to the half-open interval [0, 1). Then V is a neighborhood of x in X lying in U, and it is a union of path-connected spaces having the point x is common. Step 2. We show X is semi locally simply connected. Indeed, we show that if x  X, the x has a neighborhood U such that 1 (U, x) is trivial.

is a vertex of X. Let Stx denote the union of those edges of X that have x as an end point, and let St x denote the subspace of Stx obtained by deleting all vertices other than x. (St x is called the star of x in X). The set St x is open in X, since its complement is a union of arcs and vertices. We show that 1 (St x, x) is trivial. Let f be a loop in St x based at x. Then the image set f(1) is compact, so it lies in some finite union of arcs of Stx. Any such union is homeomorphic to the union of a finite set of line segments in the plane having an end point in common. And for any loop in such a space, the straight-line homotopy will shrink it to the constant loop at x. Theorem 1 :- Let p : E  X be a covering map, where X is a linear graph. If A is an edge of X and B is a path component of p–1 (A), then p maps B homeomorphically onto A. Furthermore, the space E is a linear graph, with the path components of the spaces p–1 (A) as its edges. Proof. Step 1. We show that p maps B homeomorphically onto A. Because the arc A is path connected and locally path connected. Since the map p0 : B  A obtained by restricting p is a covering map. Because B is path connected, the lifting correspondence  : 1 (A, a)  10p (a) is surjective; because A is simply connected, 10p (a) consists of a single point. Hence p0 is a homeomorphism. Step 2. Since X is the union of the arcs A, the space E is the union of the arcs B that are the path

respectively, with B  B'. We show B and B' intersect in at most a common end point. If A and A are equal, then B and B' are disjoint, and if A and A are disjoint, so are B and B'. Therefore, if B and B' intersect, A and A must intersect in an end point x of each; then B  B' consists of a single point, which must be an end point of each. Step 3. We show that E has the topology coherent with the arcs B. This is the hardest part of the proof. Let W be a subset of E such that W  E is open in B, for each arc B of E. We show that W is open in E. First, we show that p(W) is open in X. If A is an edge of X, then p(W)  A is the union of the sets p(W  B), as B ranges over all path components of p–1 (A). Each of these sets p (W  B) is open in A, because p maps B homeomorphically onto A; hence their union p(W)  A is open in A. Because X has the topology coherent with the subspaces A, the set p(W) is open in X. Second, we prove our result in the special case where the set W is contained in one of the slices V of p–1 (U), where U is an open set of X that is evenly covered by p. By the result just proved, we know that the set p(W) is open in X. It follows that p(W) is open in U. Because the map of V onto U obtained by restricting p is a homeomorphism, W must be open in V, and hence open in E. Finally, we prove our result in general. Choose a covering A of X by open sets U that are evenly covered by p. Then the slices V of the sets p–1 (U), for U  A, cover E. For each such slice V, let WV = WV. The set WV has the property that for each arc B of E, the set WV  B is open in B, for WV  B = (W  B)  (V  B) and both W  B and V  B are open in B. The result of the preceding paragraph implies that WV is open in E. Since W is the union of the sets WV, it also is open in E.

Now we prove the basic theorem that the fundamental group of any linear graph is a free group. Henceforth we shall refer to a linear graph simply as a graph. Definition 3 An oriented edge e of a graph X is an edge of X together with an ordering of its vertices; the first is called the initial vertex, and the second, the final vertex, of e. An edge path in X is a sequence e1, ...., en of oriented edges of X such that the final vertex of ei equals the initial vertex of ei+1, for i = 1, ...., n – 1. Such an edged path is entirely specified by the sequence of vertices x0, ...., xn, = xn. Given an oriented edge e of X, let fe be the positive linear map of [0, 1] onto e; it is a path from the initial point of e to the final point of e. Then, corresponding to the edge path e1,.... en from x0 to xn, one has the actual path f = f1  (f2  (....  fn)) from x0 to xn, where if= eif; it is uniquely determined by the edge path e1, ..., en. We call it the path corresponding to the edge path e1, .... en. If the edge path is closed, then the corresponding path f is a loop. A graph X is connected if and only if every pair of vertices of X can be joined by an edge path in X. Proof. Suppose X is connected. Define x ~ y if there is an edge path in X from x to y. For any edge of X, its end points belong to the same equivalence class; let Yx denote the union of all edges whose end points are equivalent to x. Then Yx is a sub graph of X and hence is closed in X. The sub graphs Yx form a partition of X into disjoint closed subspaces; since X is connected, there must be only one such. Conversely, suppose every pair of vertices of X can be joined by an edge path. Then they can be joined by an actual path in X. Hence all the vertices of X belong to the same component of X. Since each edge is connected, it also belongs to this component. Thus X is connected. Definition 4:- Let e1, .... en be an edge path in the linear graph X. It can happen that for some i, the oriented edges ei and ei+1 consist of the same edge of X, but with opposite orientations. If this situation does not occur, then the edge path is said to be a reduced edge path. Note that if this situation does occur, then one can delete ei and ei+1 from the sequence of oriented edges and still have an edge path remaining (provided the original sequence consists of at least three edges). This deletion process is called reducing the edge path. It enables one to show that in any connected graph, every pair of distinct vertices can be joined by a reduced edge path.

Definition 5: A sub graph T of X is said to be a tree in X if T is connected and T contains no closed reduced edge paths. A linear graph consisting of a single edge is a tree. The graph is Figure.2 is not a tree, but deletion of the edge e would make it a free. The graph in Figure 3 is a tree; deletion of the edge A would leave a tree remaining.

Theorem 2 :- Any tree T is simply connected. Proof. We first consider the case where T is a finite tree. If T consists of a single edge, then T is simply connected. If T has n edges with n > 1, there is an edge A of T such that T = T0  A, where T0 is a tree with n –1 edges and T0  A is a single vertex. Then T0 is a deformation retracts of T. Since T0 is simply connected by the induction hypothesis, so is T. To prove the general case, let f be a loop in T. The image set of f is compact and connected, so it is contained in a finite connected sub graph Y of T. Now Y contains no closed reduced edge paths, because T contains none. Thus Y is a tree. Since Y is finite, it is simply connected. Hence f is path homotopic to a constant in Y. Theorem 3 :- Let X be a connected graph. A tree T in X is maximal if and only if it contains all the vertices of X. Proof. Suppose T is a tree in X that contains all the vertices of X. If Y is a subgraph of X that properly contains T, we show that Y contains a closed reduced edge path; it follows that T is maximal. Let A be an edge of Y that is not in T; by hypothesis, the end points a and b of A belong to T. Since T is connected, we can choose a reduced edge path e1, ...., en in T from a to b. If we follow this sequence by the edge A, oriented from b to a, we obtain closed reduced edge path in Y. Now let T be a free in X that does not contain all the vertices of X. We show T is not maximal. Let x0 be a vertex of X not in T. Since X is connected, we may choose an edge path in X from x0 to a vertex of T., specified by the sequence of vertices x0, ...., xn. Let i be the smallest index such that xi  T. Let A be the edge of X with vertices xi–1 and xi. Then T  A is a tree in X, by the preceding lemma, and T  A properly contains T. Theorem 4 :- If X is a linear graph, every tree T0 in X is contained in a maximal tree in X. Proof. We apply Zorn's lemma to the collection T of all trees in X that contains T0, strictly partially ordered by proper inclusion. To show this collection has a maximal element, we need only prove the following: If T ' is a sub collection of T that is simply ordered by proper inclusion, then the union Y of the elements of T ' is a tree in X. To begin, we note that since Y is a union of sub graphs of X, it is a sub graph of X. Second, since Y is a union of connected spaces that contain the connected space T0, the space Y is connected. Finally, we suppose that e1, ...., en is a closed reduced edge path in Y and derive a contradict in. For each i, choose an element Ti of T ' that contains ei. Because T ' is simply ordered by proper inclusion, one of the trees T1, .., Tn, say Tj, contains all the others. But then e1, ...., en is a closed reduced edge path in Tj, contrary to hypothesis.

We now prove our main theorem, to the effect that a subgroup H of a free group F is free. The method of proof, remarkably enough, will give us some information about the cardinality of a system of free generator for H, when the cardinality of a system of free generators for F is known. Theorem 5:- If H is a subgroup of a free group F, then H is free. Proof. Let { |   J} be a system of free generators for F. Let X be a wedge of circles S, one for each   J; let x0 be their common point. We can give X the structure of a linear graph by breaking each circle S into three arcs, two of which have x0 as an end point. The function that assigns to each , a loop generating 1 (S, x0), induces an isomorphism of F with 1 (X, x0). Therefore we may as well assume that F equals the group 1 (X, x0). The space X is path connected, locally path connected, and semi locally simply connected. Therefore applies to show that there exists a path- connected covering space p : E  X of X such that, for some point e0 of p–1 (x0). p (1 (E, e0)) = H. Since p is a monomorphism, 1 (E, e0) is isomorphic to H. The space E is a linear graph by. Then it implies that its fundamental group is a free group. Definition 7 :- If X is a finite linear graph, we define the Euler number of X to be the number of vertices of X minus the number of edges. It is commonly denoted by the Greek letter chi, as  (X). Lemma 4 :- If X is a finite, connected linear graph, then the cardinality of a system of free generators for the fundamental group of X is 1 –  (X).

Step 1. We first show that for any finite tree T, we have (T)= 1. We proceed by induction on the number n of edges in T. If n = 1, then T has one edge and two vertices, so  (T) = 1. If n > 1, we can write T = T0  A, where T0 is a tree having n – 1 edges, and A is an edge that intersects T0 in a single vertex. We have  (T0) = 1 by the induction hypothesis. The graph T has one more edge and one more vertex than T0; hence  (T) =  (T0). Step 2. We prove the theorem. Given X, let T be an maximal tree in X. If X = T, we are finished. and T have exactly the same vertex set, and X has n more edges than T. Hence, X (X) =  (T) – n = 1 – n. So that n = 1 – X (X) Definition 8 :- Let H be a subgroup of the group G. If the collection G/H of right cosets of H in G is finite, its cardinality is called the index of H in G. (The collection of left cosets of H in G has the same cardinality, of curse). Theorem 6 :- Let F be a free group with n + 1 free generators; let H be a subgroup of F. If H has index k in F, then H has kn + 1 free generators. Proof. We apply the construction given in the proof of previous theorem. We can assume that F = 1 (X, x0), where X is a linear graph whose underlying space is a wedge of n + 1 circles. Given H, we choose a path-connected covering space p : E  X such that p (1 (E, e0)) = H. Now the lifting correspondence.  : 1 (X, x0) / H  p–1 (x0) is a bijection. Therefore, E is a k-fold covering of X. The space E is also a linear graph. Given an edge A of X, the path components of p–1 (A) are edges of E, and each is mapped by p homeomorphically onto A. Thus E has k times as many edges as X, and k times as many vertices. It follows that  (E) = k  (X). Since the fundamental group of X has n + 1 free generators, the preceding lemma tells us that  (X) = –n. Then the number of free generators of the fundamental group of E, which is isomorphic to H, is 1 –  (E) = 1 – k  (X) = 1 + kn.

1. Mac Lane S. (1963) Homology New York, Berlin, Gottingen and Heideberg: Springer, 2. Cartan, H. and S. Eilenberg, (1956) Homological Algebra. Princeton; Princeton University Press. 3. Auslander, M., and D.A Buchsoaum, "Homological dimension in Noetherian Rings II". Trans. Amer, Math. Soc. Vol. 88. (1958) pp. 194-206. 4. Bass, H. (1960) "Finitistic dimension and a homological generalisation of semi-primary 5. D.W. Hall and G.L. Spencer. (1955) Elementary Topology, John Wiley & Sons. Inc. New York. 6. J.L. Kelley, (1961) General Topology Springer-Verlag, New York. 7. D. Montgomery and L. Zippin. (1955) Topological Transformation Groups. Inter Science Publishers. Inc. New York. 8. James R. Munkres. Topology Low Price Edition. 9. G.F. Simmons. Introduction to Topology and Modern Analysis. Tata Mc. Grow Hill Edition. 10. Dr. K.K. Jha, Advanced General Topology. Nav Bharat Publication Delhi-6. 11. Alexandroff, P., (1956) Combinatorial Topology, Vol. I. Rochester N.Y. Graylock. 12. Hilton, P.J., (1953) An Introduction to Homotopy theory Cambridge. Cambridge University Press. 13. Hurewiez, S.T. (1959) Homotopy Theory, New York : Academic Press. 14. Kelley J.L. (1955) General Topology, New York; Van Nostrand. 15. Eilenberg, S. (1944) "Singular homology theory" Ann. Math 45, pp. 407-447.

Research Scholar, P.G. Department of Mathematics, Patna University, Patna, Bihar