FUNDAMENTAL GROUP OF TORUS HOMOTOPY FUNDAMENTAL GROUP OF

HOMOTOPY The torus and the mug are homotopic to each other

HOMOTOPY �Two functions in a topological space can be said to be homotopic when

HOMOTOPY �If objects have same structure, they can be classified as a class, regardless

FUNDAMENTAL GROUP �The fundamental group of an arc-wise connected set X is a group

FUNDAMENTAL GROUP OF TORUS �Product of fundamental group of circle with itself �Representation :

FUNDAMENTAL GROUP OF TORUS �IDENTITY ELEMENT: Constant path �CLOSURE: f: X Y, g: Y

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FUNDAMENTAL GROUP OF TORUS • HOMOTOPY • FUNDAMENTAL GROUP OF TORUS

HOMOTOPY

HOMOTOPY The torus and the mug are homotopic to each other

HOMOTOPY �Two functions in a topological space can be said to be homotopic when one can be continuously transformed into the other. �It is an equivalence relation. �Internal closed paths never change

HOMOTOPY �If objects have same structure, they can be classified as a class, regardless of their size, shape and dimension. �Fundamental group never changes with dilation or retraction �Fundamental group – characterizes only loops

FUNDAMENTAL GROUP �The fundamental group of an arc-wise connected set X is a group formed by sets of equivalent classes of set of all loops �Never needs a specific base point �Representation: π_1(S)

FUNDAMENTAL GROUP OF TORUS �Product of fundamental group of circle with itself �Representation : π_1(T^2)= π(S 1) x π(S 1) �Reason for not considering specific base point: Existence of homotopy between loops of torus

FUNDAMENTAL GROUP OF TORUS

FUNDAMENTAL GROUP OF TORUS �IDENTITY ELEMENT: Constant path �CLOSURE: f: X Y, g: Y Z, then f*g: X Z �ASSOCIATIVITY: f*(g*h) = (f*g)*h f: X Y g: Y �INVERSE: f: X Z h: Z Y, g: Y W X, f*g is constant path

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