Fundamental Group

9 Fundamental Group

9.1 Homotopy of paths

9.1.1 f is homotopic to f’ (two continuous maps f, f’ of the space X into the space Y) : Let I = [0,1]. there is a continuous map F : X \times I -> Y s.t. F(x,0) = f(x) and F(x,1) = f’(x).

9.1.1.1 Homotopy between f and f’ : the map F

9.1.1.2 f \simeq f’ : f is homotopic to f’

9.1.1.3 nulhomotopic f : If f \simeq f’ and f’ is a constant map

9.1.2 Path in X from x_{0} to x_{1} ( a continuous map f : [0,1] -> X ) : f(0) = x_{0} , f(1) = x_{1}

9.1.2.1 Initial point : x_{0}

9.1.2.2 Final point : x_[1]

9.1.3 Path homotopic ( two paths f, f’ : [0,1] ->X ) : They have the same initial point x_{0} and the same final point x_{1} and there is a continuous map F : I \times I -> X for each s \in I and each t \in I s.t.

9.1.3.1 F (s, 0) = f(s) and F (s, 1) = f’(s)

9.1.3.2 F (0,t) = x_{0} and F (1, t) = x_{1}

9.1.3.2.1 F : a path homotopy between f and f’

9.1.3.2.2 f \simeq_{p} f’ : f is path homotopic to f’

9.1.4 (Lem 51.1) The relations \simeq and \simeq_{p} are equivalence relations.

9.1.5 [ f ] : path-homotopy equivalence class of a path f

9.1.6 Example

9.1.6.1 Straight-line homotopy ( two maps f, g X -> \mathbb{R}^{2}) : F(x,t) = (1-t) f(x) + t g(x)

9.1.6.2 Let A be any convex subspace of \mathbb{R}^{n}. Then any two paths f, g in A from x_{0} to x_{1} are path homotopic in A.

9.1.7 Product f * g of f and g ( a path f in X from x_{0} to x_{1}, a path g in X from x_{1} to x_{2} ) : a path h given by the equations

9.1.7.1 h(s) = \begin{cases} f(2s) for s \in [0,frac{1}{2}] , \ g(2s – 1) for s \in [frac{1}{2}, 1] \end{cases}

9.1.7.2 The function h is well-defined and continuous.

9.1.7.2.1 Pf) pasting lemma

9.1.7.3 h is a path in X from x_{0} to x_{2}

9.1.8 (Groupoid properties of * )(Thm 51.2) The operation * has the following properties.

9.1.8.1 (Associativity) If [ f ] * ( [ g ] * [ h ] ) is defined, so is ( [ f ] * [ g ] ) * [ h ] ,and they are equal.

9.1.8.2 (Right and left identities) Given x \in X, let e_{x} denote the constant path e_{x} : I -> X carrying all of I to the point x. If f is a path in X from x_{0} to x_{1}, then [ f ] * [ e_{x_{1}} ] = [ f ] and [ e_{x_{0}} ] * [ f ] = [ f ]

9.1.8.3 (Inverse) Given the path f in X from x_{0} to x_{1}, let \bar{f} be the path defined by \bar{f} (s) = f (1-s) . It is called the reverse of f. Then [ f ] * [ \bar{f} ] = [ e_{x_{0}} ] and [ \bar{f} ] * [ f ] = [ e_{x_{1}} ]

9.1.8.4 Pf) convex

9.1.9 (Thm 51.3) Let f be a path in X, and let a_{0}, …, a_{n} be numbers s.t. 0 = a_{0} < a_{1} < … < a_{n} = 1. Let f_{i} : I -> X be the path that equals the positive linear map of I onto [a_{i-1}, a_{i}] followed by f. Then [ f ] = [ f_{1} ] * .. * [ f_{n} ]

9.2 Fundamental Group

9.2.1 Suppose G and G’ are groups, written multiplicatively.

9.2.1.1 Homomorphism (a map f : G -> G’) : f (x \cdot y) = f (x) \cdot f (y).

9.2.1.1.1 f( e ) = e’

9.2.1.1.2 f ( x^{-1} ) = f ( x ) ^ {-1}

9.2.1.1.3 Kernel of f : the set f^{-1} (e’) , which is subgroup of G

9.2.1.1.4 Monomorphism : an injective homomorphism

9.2.1.1.5 Epimorphism : a surjective homomorphism

9.2.1.1.6 Isomorphism : a bijective homomorphism

9.2.1.2 Left coset xH of H in G (a subgroup H of a group G ) : the set of all products xh , for h \in H

9.2.1.3 Normal subgroup H of G (a subgroup H of a group G ) : x \cdots h \cdots x^{-1} \in H for each x \in G and each h \in H.

9.2.1.4 G/H : a partition which is the collection of all coset forms.

9.2.2 Quotient of G by H (a subgroup H of a group G ) : a group (G/H, \cdot ) which \cdot is a well-defined operation (x H) \cdot (y H) = (x \cdot y) H.

9.2.2.1 If f : G -> G’ is an epimorphism, its kernel N is a normal subgroup of G and f induces an isomorphism G/N -> G’ ; x N -> f(x) for each x \in G.

9.2.3 Loop based at x_{0} ( a point x_{0} of a space X ) : a path in X that begins and ends at x_{0}

9.2.4 Fundamental group of X relative to the base point x_{0} ( a point x_{0} of a space X ) : The set of path homotopy classes of loops based at x_{0}

9.2.4.1 \pi_{1} (X, x_{0}) <- Fundamental group of X relative to the base point x_{0}

9.2.4.2 The operation * restricted to this set satisfies axioms for a group.

9.2.4.3 Also called ‘ First homotopy group of X ‘

9.2.4.4 Example

9.2.4.4.1 For a convex subset X of \mathbb{R}^{n} \pi_{1} (X, x_{0}) is trivial group.

9.2.4.4.2 Unit ball B^{n} in \mathbb{R}^{n} has trivial fundamental group.

9.2.5 \alpha – hat : let \alpha be a path in X from x_{0} to x_{1}. A map \alpha – hat is \hat{\alpha} : \pi_{1} (X, x_{0}) -> \pi_{1} (X, x_{1}) by the equation \hat{\alpha}([ f ]) – [ \bar{\alpha} ] * [ f ] * [ \alpha ].

9.2.6 (Thm 52.1) The map \hat{\alpha} is a group isomorphism.

9.2.6.1 (Cor 52.2) If X is path connected and x_{0} and x_{1} are two points of X, then \pi_{1} (X, x_{0}) is isomorphic to \pi_{1} (X, x_{1})

9.2.7 Simply connected ( a space X) : X is a path-connected space and if \pi_{1} (X, x_{0}) is the trivial (one-element) group for some x_{0} \in X, and hence for every x_{0} \in X.

9.2.7.1 \pi_{1} (X, x_{0}) : \pi_{1} (X, x_{0}) is the trivial group.

9.2.8 (Lem 52.3) In a simply connected space X, any two paths having the same initial and final points are path homotopic.

9.2.9 h : (X, x_{0}) -> (Y, y_{0}) <- a continuous map that carries the point x_{0} of X to the point y_{0} of Y

9.2.10 Homomorphism h_{} induced by h relative to the base point x_{0} (a continuous map h : (X, x_{0}) -> (Y, y_{0}) ) : h_{} : \pi_{1} (X, x_{0}) -> \pi_{1} (Y, y_{0}) by the equation h_{*} ([ f ]) = [h \bullet f ]

9.2.11 (Thm 52.4) If h : (X, x_{0}) -> (Y, y_{0}) and k : (Y, y_{0}) -> (Z, z_{0}) are continuous, then (k \bullet h){*} = k{} \bullet h_{} . If i: (X, x_{0}) -> (X, x_{0}) is the identity map, then i_{*} is the identity homomorphism.

9.2.11.1 (Cor) If h : (X, x_{0}) -> (Y, y_{0}) is a homomorphism of X with Y, then h_{*} is an isomorphism of \pi_{1} (X, x_{0}) with \pi_{1} (Y, y_{0})

9.3 Covering Spaces

9.3.1 Evenly covered by p ( an open set U of B, a map p : E -> B ) : Let p be continuous surjective map. The inverse image p^{-1} (U) can be written as the union of disjoint open sets V_{\alpha} in E s.t. for each \alpha, the restriction p to V_{\alpha} is a homeomorphism of V_{\alpha} onto U.

9.3.1.1 A partition of p^{-1} (U) into slices : the collection {V_{\alpha}}

9.3.2 Covering map ( a continuous surjective map p : E -> B) : Every point b of B has a neighborhood U that is evenly covered by p.

9.3.2.1 a covering space E of B : if a covering map p : E -> B exists

9.3.2.2 If p : E->B is a covering map, then for each b \in B the subspace p^{-1}(b) of E has the discrete topology.

9.3.2.3 If p : E ->B is a covering map, then p is an open map.

9.3.3 (Thm 53.1) The map p : \mathbb{R} -> S^{-1} given by the equation p(x) = (cos 2 \pi x , sin 2 \pi x ) is a covering map.

9.3.4 A local homeomorphism p of E with B ( a map p : E-> B) : each point e of E has a neighborhood that is mapped homeomorphically by p onto an open subset of B.

9.3.4.1 If p: E -> B is a covering map, then p is a local homeomorphism of E with B.

9.3.4.2 Example

9.3.4.2.1 The map p : \mathbb{R}_{+} -> S^{1} ; p(x) = (cos 2 \pi x , sin 2 \pi x )

9.3.5 (Thm 53.2) : Let p : E->B be a covering map. If B_{0} is a subspace of B, and if E_{0} = p^{-1}(B_{0}), then the map p_{0} : E_{0} -> B_{0} obtained by restricting p is a covering map.

9.3.6 (Thm 53.3) If p : E -> B and p’ : E’ -> B’ are covering maps, then p \times p’ : E \times E’ -> B \times B’ is a covering map.

9.3.6.1 Example

9.3.6.1.1 For a Torus T = S^{-1} \times S^{-1}, p \times p : \mathbb{R} \times \mathbb{R} -> T

9.3.6.1.2 Let b_{0} = p(0) of S^{1}; a figure-eight space B_{0} = a subspace (S^{1} \times b_{0}) U (b_{0} \times S^{1}) of S^{1} \times S^{1} ; an infinite grid E_{0} = (\mathbb{R} \times \mathbb{Z}) \cup (\mathbb{Z} \times \mathbb{R} ). Then the map p_{0} : E_{0} -> B_{0} = p \times p | E_{0} is a covering map.

9.4 Fundamental group of the circle

9.4.1 Lifting of f ( a continuous mapping f of some space X into B) : Let p : E-.B be a map. A lifting of f is a map \tilde{f} : X -> E s.t. p \bullet \tilde{f} = f.

9.4.2 (Lem 54.1) Let p : E -> B be a covering map, let p(e_{0}) = b_{0}. Any path f : [0,1] -> B beginning at b_{0} has a unique lifting to a path \bar{f} in E beginning at e_{0}.

9.4.2.1 Pf) (Lebesgue number lemma) (pasting lemma)

9.4.3 (Lem 54.2) Let p : E->B be a covering map; let p(e_{0}) = b_{0}. Let the map F : I \times I -> B be continuous, with F (0,0) = b_{0}. There is a unique lifting of F to a continuous map \tilde{F} : I \times I -> E s.t. \tilde{F} (0,0) = e_{0}. If F is a path homotopy, then \tilde{F} is a path homotopy.

9.4.3.1 Pf) (Lem 54.1) (Lebesgue number lemma) (pasting lemma)

9.4.4 (Thm 54.3) Let p : E -> B be a covering map ; let p(e_{0}) = b_{0}. Let f and g be two paths in B from b_{0} to b_{1}; let \tilde{f} and \tilde{g} be their respective liftings to paths in E beginning ar e_{0}. If f and g are path homotopic, then \tilde{f} and \tilde{g} end at the same point of E and path homotopic.

9.4.5 Lifting correspondence \phi derived from the covering map p ( a covering map p: E -> B) : Let b_{0} \in B. Choose e_{0} so that p(e_{0}} = b_{0}. Given an element [ f ] of \pi_{1} (B,b_{0}) , let \tilde{F} be the lifting of f to a path in E that begins at e_{0}. Let \phi([ f ]) denote the end point \tilde{f} (1) of \tilde{f}. Then \phi is a well-defined set map \phi : \pi_{1} (B,b_{0}) -> p^{-1} (b_{0}).

9.4.6 (Thm 54.4) Let p: E->B be a covering map; let p(e_{0}) = b_{0}. If E is a path connected, then the lifting correspondence \phi : \pi_{1} (B,b_{0}) -> p^{-1} (b_{0}). Is surjective. If E is simply connected, it is bijective.

9.4.7 (Thm 54.5) The fundamental group of S^{1} is isomorphic to the additive group of integers.

9.4.7.1 Pf) (Thm 53.1)

9.4.8 Cyclic group (a group G) : Let x be an element of G. the set of all elements of the form x^{m}, for m \in \mathbb{Z}, equals G

9.4.8.1 The symbol x^{n} denotes the n-fold product of x with itself, x^{-n} denotes the n-fold product of x^{-1} with itself, and x^{0} denotes the identity element of G.

9.4.8.2 x : a generator of G

9.4.9 Order of the group : Cardinality of a group

9.4.10 (Thm 54.6) Let p : E->B be a covering map ; let p(e_{0}) = b_{0}.

9.4.10.1 The homomorphism p_{*} : \pi_{1} (E,e_{0}) -> \pi_{1} (B,b_{0}) is a monomorphism.

9.4.10.2 Let H = p_{*}( \pi_{1} (E,e_{0}) ). The lifting correspondence \phi induces an injective map \Phi : \pi_{1} (B,b_{0}) / H -> p^{-1} (b_{0}) of the collection of right cosets of H into p^{-1} (b_{0}), which is bijective if E is path connected.

9.4.10.3 If f is a loop in B based at b_{0}, then [ f ] \in H iff f lifts to a loop in E based at e_{0}.

9.4.10.3.1 Pf) \phi ( [ f ] ) = \phi ( [ g ] ) iff [ f ] \in H * [ g ].

9.5 Retractions and Fixed Points

9.5.1 Retraction of X onto A ( a subset A of X ) : a continuous map r : X -> A s.t. r |A is the identity map of A.

9.5.1.1 Retract A of X : If such a map r exists

9.5.2 (Lem 55.1) If A is a retract of X, then the homomorphism of fundamental groups induced by inclusion j : A ->X is injective.

9.5.3 (No-retraction theorem) (Thm 55.2) : There is no retraction of B^{2} onto S^{1}.

9.5.3.1 B^{2} is a unit ball.

9.5.4 (Lem 55.3) Let h : S^{1} -> X be a continuous map. Then the following conditions are equivalent.

9.5.4.1 h is nullhomotopic.

9.5.4.2 h extends to a continuous map k : B^{2} -> X.

9.5.4.3 h_{*} is the trivial homomorphism of fundamental groups.

9.5.5 (Cor 55.4) The inclusion map j : S^{1} -> \mathbb{R}^{2} - \mathbf{0} is not nullhomotopic. The identity map I : S^{1} -> S^{1} is not nullhomotopic.

9.5.6 (Thm 55.5) Given a nonvanishing vector field on B^{2}, there exists a point of S^{1} where the vector field points directly inward and a point of s^{1} where it points directly outward.

9.5.6.1 Vector field on B^{2} : an ordered pair (x, v(x)), where x is in B^{2} and v is a continuous map of B^{2} into \mathbb{R}^{2}.

9.5.6.2 Nonvanishing (a vector field) : v(x) \neq \mathbf{0} for every x

9.5.6.3 Pf) (Cor 55.4)

9.5.7 (Brower fixed-point theorem for the disc) (Thm 55.6) : If f : B^{2} -> B^{2} is continuous, then there exists a point x \in B^{2} s.t. f(x) = x

9.5.7.1 (Cor 55.7) Let A be a 3 by 3 matrix of positive real numbers. Then A has a positive real eigenvalue.

9.5.8 (Thm 55.8) There is an \epsilon > 0 s.t. for every open covering \mathcal{A} of T by sets of diameter less than \epsilon, some point of T belongs to at least three elements of \mathcal{A}.

9.5.8.1 T = {(x,y) | x \ge 0 and y \ge 0 and x + y \le 1}

9.5.8.2 Pf) T is homeomorphic to B^{2}, a partition of unity,

9.6 Fundamental Theorem of Algebra

9.6.1 (Fundamental theorem of algebra) (Thm 56.1) A polynomial equation x^{n} + a_{n-1} x^{n-1} + … + a_{1}x + a_{0} = 0 of degree n > 0 with real or complex coefficients has at least one (real or complex) root.

9.6.1.1 Pf) g: S^{1} -> \mathbb{R}^{2} - \mathbf{0} ; g(z) = z^{n} is not nullhomotopic.

9.7 Borsuk-Ulam Theorem

9.7.1 Antipode (a point x in S^{n}) : -x

9.7.2 Antipode-preserving h ( a map h : S^{n} -> S^{m}) : h(-x) = -h(x) for all x \in S^{n}.

9.7.3 (Thm 57.1) if h : S^{1} -> S^{1} is continuous and antipode-preserving, then h is not nullhomotopic.

9.7.3.1 Pf) covering map,

9.7.4 (Thm 57.2) There is no continuous antipode-preserving map g : S^{2} -> S^{1}.

9.7.4.1 Pf) (Thm 57.1)

9.7.5 (Borsuk-Ulam theorem for S^{2}) (Thm 57.3) Given a continuous map f : S^{2} -> \mathbb{R}^{2}, there is a point x of S^{2} s.t. f(x) = f(-x).

9.7.6 (The bisection theorem) (Thm 57.4) Given two bounded polygonal regions in \mathbb{R}^{2}, there exists a line in \mathbb{R}^{2} that bisects each of them.

9.7.6.1 Pf) (Borsuk-Ulam theorem)

9.8 Deformation Retracts and Homotopy type

9.8.1 (Lem 58.1) Let h, k : (X, x_{0}) -> (Y, y_{0}) be continuous maps, If h and k are homotopic, and if the image of the base point x_{0} of X remains fixed at y_{0} during the homotopy, then the homomorphisms h_{} and k_{} are equal.

9.8.2 (Thm 58.2) The inclusion map j : S^{n} -> \mathbb{R}^{n+1} - \mathbf{0} induces an isomorphism of fundamental groups.

9.8.2.1 Pf) (Lem 58.1)

9.8.3 Deformation retract of X ( a subspace A of X) : the identity map of X is homotopic to a map that carries all of X into A, such that each point of A remains fixed during the homotopy.

9.8.3.1 Deformation retraction H of X onto A ( a homotopy H) : There is a continuous map H : X \times I -> X s.t. H(x,0) = x and H(x,1) \in A for all x \in X, and H(a,t) = a for all a \in A.

9.8.3.2 The map r : X -> A defined by the equation r(x) = H(x,1) is a retraction of X onto A.

9.8.3.3 H is a homotopy between the identity map of X and the map j \bullet r, where j : A -> X is inclusion.

9.8.4 (Thm 58.3) Let A be a deformation retract of X ; let x_{0} \in A. Then the inclusion map j : (A, x_{0}) -> (X, x_{0}) induces an isomorphism of fundamental groups.

9.8.5 Homotopy equivalences ( continuous maps f : X -> Y, g : Y -> X) : the map g \bullet f : X -> X is homotopic to the identity map of X, and the map f \bullet g : Y -> Y is homotopic to the identity map of Y.

9.8.5.1 Each is said to be a ‘homotopy inverse’ of the other.

9.8.5.2 Relation of homotopy equivalence is an equivalence relation.

9.8.5.3 Same Homotopy type : Spaces that are homotopy equivalent

9.8.6 (Lem 58.4) Let h, k : X -> Y be continuous maps; let h(x_{0}) = y_{0} and k(x_{0}) = y_{1}. If h and k are homotopic, there is a path \alpha in Y from y_{0} to y_{1} s.t. k_{} = \hat{\alpha} \bullet h_{}. Indeed, if H : X \times I -> Y is the homotopy between h and k, then \alpha is the path \alpha(t) = H(x_{0}, t)

9.8.6.1 (Cor 58.5) Let h, k : X ->Y be homotopic continuous maps; let h(x_{0}) = y_{0} and k(x_{0}) = y_{1}. If h_{} is injective, or surjective, or trivial, so is k_{}.

9.8.6.2 (Cor 58.6) Let h : X -> Y. If h is nullhomotopic, then h_{*} is the trivial homomorphism.

9.8.7 (Thm 58.7) Let f : X -> Y be continuous; let f(x_{0}) = y_{0}. If f is a homotopy equivalence, then f_{*} : \pi_{1} (X, x_{0}) ->\pi_{1} (Y, y_{0}) is an isomorphism.

9.9 Fundamental Group of S^{n}

9.9.1 (Thm 59.1) Suppose X = U \cup V, where U and V are open sets of X. Suppose that U \cap V is path connected, and that x_{0} \in U \cap V. Let I and j be the inclusion mappings of U and V, respectively, into X. Then the images of the induced homomorphisms i_{} : \pi_{1} (U, x_{0}) ->\pi_{1} (X, x_{0}) and j_{} : \pi_{1} (V, x_{0}) ->\pi_{1} (X, x_{0}) generate \pi_{1} (X, x_{0})

9.9.1.1 Pf) there is a subdivision a_{0} < a_{1} < … < a_{n} of the unit interval s.t. f(a_{i}) \in U \cap V and f([a_{i-1}, a_{i}]) is contained either in U or in V, for each i., (Lebesgue number lemma)

9.9.1.2 (Cor 59.2 ) Suppose X = U \cup V, where U and V are open sets of X; suppose X \cap V is nonempty and path connected, If U and V are simply connected, then X is simply connected.

9.9.2 (Thm 59.3) If n \ge 2, then n-sphere S^{n} is simply connected.

9.9.2.1 Stereographic projection f : (S^{n} – p) -> \mathbb{R}^{n} ; f(x) = f(x_{1}, …, x_{n+1}) = frac{1}{1-x_{n+1}} (x_{1}, …, x_{n}) where p = (0,…,0,1) \in \mathbb{R}^{n+1}

9.9.2.2 Pf) (Cor 59.2)

9.10 Fundamental Groups of Some Surfaces

9.10.1 (Thm 60.1) \pi_{1} (X \times Y, x_{0} \times y_{0}) is isomorphic with \pi_{1}(X,x_{0}) \times \pi_{1} (Y, y_{0}).

9.10.1.1 (Cor 60.2) The fundamental group of the torus T = S^{1} \times S^{1} is isomorphic to the group \mathbb{Z} \times \mathbb{Z}.

9.10.2 Projective plane P^{2} : quotient space obtained from S^{2} by identifying each point x of S^{2} with its antipodal point -x.

9.10.3 (Thm 60.3) The projective plane P^{2} is a compact surface, and the quotient map p: S^{2} -> P^{2} is a covering map.

9.10.3.1 Pf) S^{2} is normal and p is a closed map -> P^{2} is Hausdorff

9.10.3.2 (Cor 60.4) \pi_{1} (P^{2}, y) is a group of order 2.

9.10.3.2.1 Pf) (Thm 54.4)

9.10.4 Projective n-space : quotient space obtained from S^{n} by identifying each point x of S^{2} with its antipodal point -x

9.10.4.1 The projective plane P^{n} is a compact surface, and the quotient map p: S^{n} -> P^{n} is a covering map.

9.10.5 (Lem 60.5) The fundamental group of the figure eight is not abelian.

9.10.6 (Thm 60.6) The fundamental group of the double torus is not abelian.

9.10.6.1 Double torus T#T : the surface obtained by taking two copies of the torus, deleting a small open disc from each of them, and pasting the remaining pieces together along their edges.

9.10.6.2 Pf) Figure eight X is a retract of T#T.

9.10.6.3 (Cor 60.7) The 2-sphere, torus, projective plane and double torus are topologically distinct.