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A Very Brief Introduction to Homotopy by Juliet Cooke

As part of her final year project, Juliet Cooke has written a brief introduction to homotopy, one of the key concepts in algebraic topology. She has agreed to allow us to make this available on the MathSoc website under the Creative Commons Attribution License.

You can also download the article in full pdf form here.

In this article I shall be considering the notion of homotopy equivalence, which plays a central role in Algebraic Topology.

Homotopy Equivalence of Functions

contour

Two contours C_1 and C_2 for which the line integrals are equal. The points in red mark singularities of the integrand.

When evaluating complex contour integrals the exact shape of the contour doesn't matter as one can 'continuously deform' the contour without changing the value of the integral so long as the 'continuous deformation' does not pass over any singularities of the integrand. This significantly simplifies the complex integration of relatively well behaved complex functions and allows one to prove Cauchy's theorem and the residue theorem.

Definition (Homotopy equivalence)

Let X and Y be two topological spaces with continuous maps f : X \to Y and g: X \to Y between them. Then a homotopy between f and g is a continuous map H : X \times [0, 1] \to Y such that for all x \in X,

  • H(x, 0) = f and
  • H(x, 1) = g.

If such a homotopy H exists then f and g are said to be homotopy equivalent or homotopic. This is denoted H: f \simeq g or simply f \simeq g when this does not lead to ambiguity.

As is betrayed by the name, homotopy equivalence is an equivalence relation. Furthermore by thinking of contours as continuous maps from [0, 1] to \mathbb{C} we now have exactly the equivalence which matters in contour integration. So we can now reformulate the first paragraph precisely in the following Proposition.

Proposition

Let f: \mathbb{C} \to \mathbb{C} be a holomorphic function on the open set A \subseteq \mathbb{C}. Then, if C_1, C_2 : [0, 1] \to A are two homotopy equivalent contours,

 \int_{C_1} f = \int_{C_2} f.

Homotopy Equivalence of Spaces

Two shapes that seem very similar.

Homotopy is not restricted to maps between spaces but can also be used to formulate a enlightening form of equivalence between topological spaces themselves. To illustrate this consider the two shapes in the above image (this example is from and elaborated on in Hatcher). It is clear that they have the same essential shape as they are both used to represent the same letter in English. However they are not homeomorphic as there exists no invertible map between them.

Definition (Homotopy equivalence of spaces)

Two spaces X and Y are \textbf{homotopy equivalent} if there is a homotopy equivalence f : X \to Y between them. f : X \to Y is a homotopy equivalence if there exists a map g : Y \to X such that gf \simeq 1_X and fg \simeq 1_Y.

However using this homotopy equivalence the As would also be homotopic to the O below. The two thick letters are even homeomorphic. However if we superimpose the images and then consider deformation retraction there is indeed a difference between the As and the Os which none the less makes the two As equivalent.

O

This O is homotopic to the As. So how does it differ?

AO

When superimposed the difference is clear, O is not even a subset of A.

Definition (Deformation retraction (Hatcher))

A retract between from a space X to a subspace A is a continuous map r: X \to A such that r|_A = 1_A, or equivalently r^2 = r. A deformation retraction is a homotopy H: 1_X \simeq r relative to A, that is H(A, t) = 1_A for all t \in [0, 1]. In this case we say that X deformation retracts onto A.

As can be seen in the figure above the O is not even a subset of the thick A thus the A certainly does not deform retract onto O or vice versa. However, the thick A does deform retract onto the thin A. This deformation retraction shrinks the width of the thick A and is fixed on the thin A throughout.

Homotopy Groups

The homotopic equivalence did however capture that both the As and the O have one hole in them. Holes in topological spaces can be detected by considered algebraic structures induced by them. In particular the fundamental group and higher homotopy groups (or homology groups which will not be covered here).

Fundamental Group (Strom)

Let X be a topological space. The fundamental group of X with respect to basepoint x \in X is,

\pi_1 (X, x) = \left[ S^1, X \right].


Where, if S^1 is given a standard basepoint \star \in S^1 then, \left[ S^1, X \right] is the set of homotopy classes of continuous maps f: S^1 \to X
which preserve basepoint (f(\star) = x).

A loop in O with winding number 2.

A loop in O with winding number 2.

It can be shown that the fundamental group is in fact a group under concatenation of paths. One can also show that the homotopy classes of a loop in O can be determined entirely by the loops winding number. Thus for any x \in O we have that \pi_1(0, x) \cong \mathbb{Z}.

Two spaces that need the 2nd homotopy group to distinguish between.

Two spaces that need the 2^{nd} homotopy group to distinguish between.

However the fundamental group cannot distinguish between D^3 - \frac{1}{2}D^3 and D^3 (shown above) as in both spaces loops are homotopy equivalent to a constant loop at basepoint. To do this we need to introduced the higher homotopy groups.

Definition (Higher homotopy groups (Strom))

Let X be a topological space with a point x \in X. Then the n^{th} homotopy group of X at x is,

\pi_n(X, x) = \left[ S^n, X \right].

The `two dimensional' hole in D^3 - \frac{1}{2}D^3 can now be detected by finding that \pi_2(D^3 - \frac{1}{2}D^3, x) \cong \mathbb{Z}. However, in general calculating higher homotopy groups can be very difficult. An illustration of this difficulty is that even calculating \pi_n(S^m), in general, when n data-recalc-dims=m" /> is still an open problem of homotopy theory.

References

  • (Strom) Strom, J. 2011. Modern Classical Homotopy Theory. American Mathematical Society.
  • (Hatcher) Hatcher, A. 2001. Algebraic Topology. Cambridge University Press.