12.2 Introduction to Algebraic Topology

Algebraic topology bridges the gap between algebra and topology, using algebraic tools to study topological spaces. It introduces key concepts like homotopy, homology groups, and the fundamental group to classify and distinguish spaces based on their structural properties.

These tools provide powerful ways to analyze spaces, revealing information about holes, connectedness, and other topological features. By translating geometric problems into algebraic ones, we can apply familiar algebraic techniques to solve complex topological questions.

Fundamental concepts in algebraic topology

Homotopy and continuous deformation

Top images from around the web for Homotopy and continuous deformation

• 

  Homotopy principle - Wikipedia View original

  Is this image relevant?

• 

  Homotopy of paths | TikZ example View original

  Is this image relevant?

• 

  at.algebraic topology - Determining homotopy classes [T^2, RP^2] - MathOverflow View original

  Is this image relevant?

• 

  Homotopy principle - Wikipedia View original

  Is this image relevant?

• 

  Homotopy of paths | TikZ example View original

  Is this image relevant?

1 of 3

Top images from around the web for Homotopy and continuous deformation

• 

  Homotopy principle - Wikipedia View original

  Is this image relevant?

• 

  Homotopy of paths | TikZ example View original

  Is this image relevant?

• 

  at.algebraic topology - Determining homotopy classes [T^2, RP^2] - MathOverflow View original

  Is this image relevant?

• 

  Homotopy principle - Wikipedia View original

  Is this image relevant?

• 

  Homotopy of paths | TikZ example View original

  Is this image relevant?

1 of 3

• Homotopy formalizes the notion of two maps being "the same" from a topological perspective
• Homotopy is a continuous deformation of one map into another
  • Two continuous functions $f,g:X→Y$ are homotopic if there exists a continuous function $H:X×[0,1]→Y$ such that $H(x,0)=f(x)$ and $H(x,1)=g(x)$ for all $x∈X$
• The homotopy class of a map $f:X→Y$, denoted $[f]$, is the set of all maps homotopic to $f$
  • The set of homotopy classes of maps from $X$ to $Y$ is denoted $[X,Y]$

Homology groups and topological invariants

• Homology is a functor that assigns to each topological space $X$ a sequence of abelian groups $H_n(X)$, called the homology groups of $X$
  • Homology groups encode information about the "holes" in $X$
  • The $n$-th homology group $H_n(X)$ measures the number of $n$-dimensional holes in $X$ that cannot be filled in by $(n+1)$-dimensional subspaces
• Homology groups are topological invariants
  • If $X$ and $Y$ are homeomorphic, then $H_n(X)≅H_n(Y)$ for all $n$
  • Topological invariants remain unchanged under homeomorphisms (continuous bijections with continuous inverses)

Homology groups for topological spaces

Computing homology groups for simple spaces

• The homology groups of a point are $H_n(pt)=0$ for all $n>0$ and $H_0(pt)=\mathbb{Z}$
• The homology groups of the $n$-sphere $S^n$ are:
  • $H_k(S^n)=\mathbb{Z}$ if $k=0$ or $k=n$
  • $H_k(S^n)=0$ otherwise
• The homology groups of the $n$-torus $T^n$ are $H_k(T^n)=\mathbb{Z}^{\binom{n}{k}}$ for all $k$

Techniques for computing homology groups

• The Mayer-Vietoris sequence relates the homology of a space $X$ to the homology of two subspaces $A,B⊆X$ whose union is $X$
  • Allows for the computation of homology groups by decomposing a space into simpler pieces (e.g., decomposing a torus into two cylinders)
• The Künneth formula relates the homology of a product space $X×Y$ to the homology of its factors $X$ and $Y$
  • Allows for the computation of homology groups of product spaces (e.g., computing the homology of a torus as a product of circles)

Fundamental group for classifying spaces

Definition and properties of the fundamental group

• The fundamental group $π_1(X,x_0)$ of a topological space $X$ at a basepoint $x_0∈X$ is the set of homotopy classes of loops based at $x_0$
  • The group operation is given by concatenation of loops
• The fundamental group is a topological invariant
  • If $X$ and $Y$ are homeomorphic, then $π_1(X,x_0)≅π_1(Y,y_0)$ for any choice of basepoints $x_0∈X$ and $y_0∈Y$
• Spaces with isomorphic fundamental groups are homotopy equivalent, but the converse is not true in general

Using the fundamental group to distinguish spaces

• The fundamental group can be used to distinguish between spaces that are not homeomorphic
  • The circle $S^1$ has fundamental group $π_1(S^1)≅\mathbb{Z}$
  • The punctured plane $\mathbb{R}^2 \setminus \{0\}$ has fundamental group $π_1(\mathbb{R}^2 \setminus \{0\})≅\mathbb{Z}$
  • Since $S^1$ and $\mathbb{R}^2 \setminus \{0\}$ have isomorphic fundamental groups but are not homeomorphic, the fundamental group alone is not sufficient to determine homeomorphism type
• The Seifert-van Kampen theorem allows for the computation of the fundamental group of a space by decomposing it into simpler pieces whose fundamental groups are known
  • Useful for computing fundamental groups of spaces obtained by gluing or attaching (e.g., wedge sum, connected sum)

Algebraic invariants vs topological properties

Relationship between algebraic invariants and topology

• Algebraic invariants, such as homology and homotopy groups, provide a way to study topological spaces using algebraic tools
  • Homology groups encode information about the holes in a space
  • Homotopy groups encode information about the maps into a space
• If two spaces have different algebraic invariants (e.g., non-isomorphic homology or homotopy groups), then they cannot be homeomorphic
• However, spaces with isomorphic algebraic invariants are not necessarily homeomorphic
  • There exist non-homeomorphic spaces with isomorphic homology and homotopy groups (e.g., the Poincaré homology sphere and the 3-sphere)

Topological properties defined by algebraic invariants

• Algebraic invariants can be used to define and study topological properties
• Connectedness
  • A space $X$ is path-connected if and only if $H_0(X)≅\mathbb{Z}$
  • Path-connectedness implies connectedness, but the converse is not true in general (e.g., the topologist's sine curve)
• Orientability
  • A closed $n$-manifold $M$ is orientable if and only if $H_n(M)≅\mathbb{Z}$
  • Non-orientable manifolds (e.g., the Möbius strip, the real projective plane) have torsion in their top homology group