Lecture 2: Algebraic Topology Gunnar Carlsson Stanford University I Developed starting in late 19th century I Powerful methods for understanding shape I Regarded as esoteric Algebraic topology I Formalism to \measure" aspects of shape I Powerful methods for understanding shape I Regarded as esoteric Algebraic topology I Formalism to \measure" aspects of shape I Developed starting in late 19th century I Regarded as esoteric Algebraic topology I Formalism to \measure" aspects of shape I Developed starting in late 19th century I Powerful methods for understanding shape Algebraic topology I Formalism to \measure" aspects of shape I Developed starting in late 19th century I Powerful methods for understanding shape I Regarded as esoteric Motivation I Can we count the number of distinct obstacles? I Can we describe the \shape" of the obstacles? Motivation I How can we detect the presence of obstacles in a region in the plane? I Can we describe the \shape" of the obstacles? Motivation I How can we detect the presence of obstacles in a region in the plane? I Can we count the number of distinct obstacles? Motivation I How can we detect the presence of obstacles in a region in the plane? I Can we count the number of distinct obstacles? I Can we describe the \shape" of the obstacles? I Only access we have is the ability to throw a lasso into the region and attempt to pull it tight I If I pull it tight and the whole lasso does not return to me, it is \hung up" on an obstacle Motivation I Suppose we can't see or access the region directly I If I pull it tight and the whole lasso does not return to me, it is \hung up" on an obstacle Motivation I Suppose we can't see or access the region directly I Only access we have is the ability to throw a lasso into the region and attempt to pull it tight Motivation I Suppose we can't see or access the region directly I Only access we have is the ability to throw a lasso into the region and attempt to pull it tight I If I pull it tight and the whole lasso does not return to me, it is \hung up" on an obstacle Motivation I We can't count the number of obstacles Motivation I There is an obstacle Motivation I There is an obstacle I We can't count the number of obstacles Motivation Motivation Motivation I All the experiments have identical results, so cannot count I How can we attempt to count the number of obstacles I Consider the set of all possible lasso tosses in the region I The set of such tosses is uncountably infinite Motivation I In each case, we detect the presence of an obstacle I How can we attempt to count the number of obstacles I Consider the set of all possible lasso tosses in the region I The set of such tosses is uncountably infinite Motivation I In each case, we detect the presence of an obstacle I All the experiments have identical results, so cannot count I Consider the set of all possible lasso tosses in the region I The set of such tosses is uncountably infinite Motivation I In each case, we detect the presence of an obstacle I All the experiments have identical results, so cannot count I How can we attempt to count the number of obstacles I The set of such tosses is uncountably infinite Motivation I In each case, we detect the presence of an obstacle I All the experiments have identical results, so cannot count I How can we attempt to count the number of obstacles I Consider the set of all possible lasso tosses in the region Motivation I In each case, we detect the presence of an obstacle I All the experiments have identical results, so cannot count I How can we attempt to count the number of obstacles I Consider the set of all possible lasso tosses in the region I The set of such tosses is uncountably infinite Motivation I Called homotopy I Consider the set E of equivalence classes of tosses I E turns out to be countable I Appears to be progress, but still can't count number of holes Motivation I Define an equivalence relation on the set of lasso tosses I Consider the set E of equivalence classes of tosses I E turns out to be countable I Appears to be progress, but still can't count number of holes Motivation I Define an equivalence relation on the set of lasso tosses I Called homotopy I E turns out to be countable I Appears to be progress, but still can't count number of holes Motivation I Define an equivalence relation on the set of lasso tosses I Called homotopy I Consider the set E of equivalence classes of tosses I Appears to be progress, but still can't count number of holes Motivation I Define an equivalence relation on the set of lasso tosses I Called homotopy I Consider the set E of equivalence classes of tosses I E turns out to be countable Motivation I Define an equivalence relation on the set of lasso tosses I Called homotopy I Consider the set E of equivalence classes of tosses I E turns out to be countable I Appears to be progress, but still can't count number of holes Motivation Motivation I Concatenation of lasso tosses I For this situation E turns out to be a free group on two generators I The number of generators of a finitely generated free group is a well defined invariant of the group I Solution to the counting problem Motivation I Solution: recognize that E carries a group structure I For this situation E turns out to be a free group on two generators I The number of generators of a finitely generated free group is a well defined invariant of the group I Solution to the counting problem Motivation I Solution: recognize that E carries a group structure I Concatenation of lasso tosses I The number of generators of a finitely generated free group is a well defined invariant of the group I Solution to the counting problem Motivation I Solution: recognize that E carries a group structure I Concatenation of lasso tosses I For this situation E turns out to be a free group on two generators I Solution to the counting problem Motivation I Solution: recognize that E carries a group structure I Concatenation of lasso tosses I For this situation E turns out to be a free group on two generators I The number of generators of a finitely generated free group is a well defined invariant of the group Motivation I Solution: recognize that E carries a group structure I Concatenation of lasso tosses I For this situation E turns out to be a free group on two generators I The number of generators of a finitely generated free group is a well defined invariant of the group I Solution to the counting problem I Lasso tosses should be thought of as (based) maps of circle into our region X I The homotopy equivalence relation is encoded by maps S1 × I ! X , so that the base point always goes to base point I The set of equivalence classes is denoted by π1(X ; x) I The set π1(X ; x) obtains a group structure by concatenation of loops I Called the fundamental group Homotopy Groups I Need to turn this into mathematics I The homotopy equivalence relation is encoded by maps S1 × I ! X , so that the base point always goes to base point I The set of equivalence classes is denoted by π1(X ; x) I The set π1(X ; x) obtains a group structure by concatenation of loops I Called the fundamental group Homotopy Groups I Need to turn this into mathematics I Lasso tosses should be thought of as (based) maps of circle into our region X I The set of equivalence classes is denoted by π1(X ; x) I The set π1(X ; x) obtains a group structure by concatenation of loops I Called the fundamental group Homotopy Groups I Need to turn this into mathematics I Lasso tosses should be thought of as (based) maps of circle into our region X I The homotopy equivalence relation is encoded by maps S1 × I ! X , so that the base point always goes to base point I The set π1(X ; x) obtains a group structure by concatenation of loops I Called the fundamental group Homotopy Groups I Need to turn this into mathematics I Lasso tosses should be thought of as (based) maps of circle into our region X I The homotopy equivalence relation is encoded by maps S1 × I ! X , so that the base point always goes to base point I The set of equivalence classes is denoted by π1(X ; x) I Called the fundamental group Homotopy Groups I Need to turn this into mathematics I Lasso tosses should be thought of as (based) maps of circle into our region X I The homotopy equivalence relation is encoded by maps S1 × I ! X , so that the base point always goes to base point I The set of equivalence classes is denoted by π1(X ; x) I The set π1(X ; x) obtains a group structure by concatenation of loops Homotopy Groups I Need to turn this into mathematics I Lasso tosses should be thought of as (based) maps of circle into our region X I The homotopy equivalence relation is encoded by maps S1 × I ! X , so that the base point always goes to base point I The set of equivalence classes is denoted by π1(X ; x) I The set π1(X ; x) obtains a group structure by concatenation of loops I Called the fundamental group I The case n = 0 is special, carries no group structure I π0(X ; x) as a set is the set of connected components of X I πn(X ; x) is abelian for n ≥ 2 I Although natural easy to define, very difficult to compute I Would like to find an invariant which is easier to compute, but which captures some of the same information Homotopy Groups n I Notion can be extended to maps of S for all n, to obtain πn(X ; x) I π0(X ; x) as a set is the set of connected components of X I πn(X ; x) is abelian for n ≥ 2 I Although natural easy to define, very difficult to compute I Would like to find an invariant which is easier to compute, but which captures some of the same information Homotopy Groups n I Notion