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3. Homotopy

3.1 Homotopic maps

Let X,Y be two topological spaces, and I the closed unit interval [0,1]. Two maps f,g from X to Y are called homotopic if there exists a map F from X × I to Y such that F(x,0) = f(x) and F(x,1) = g(x) for all x. Here F is called a homotopy (from f to g). Intuitively, the second argument can be viewed as time, and then the homotopy describes a continuous deformation from f to g.

A map is called nullhomotopic when it is homotopic to a constant map.

A space is called contractible when the identity map from the space to itself is nullhomotopic. For example, each real topological vector space is contractible - a suitable homotopy is given by F(x,t) = (1-t)x. More generally, each convex subset of a real topological vector space is contractible.

If Y is contractible, then each map from some topological space X to Y is nullhomotopic. Also, if X is contractible, then each map from X into some topological space Y is nullhomotopic.


(i) If Y is not pathconnected, then not all constant maps into Y are homotopic.

(ii) Let S(n) be the n-dimensional sphere (points of norm 1 in (n+1)-space). If f,g are two maps from some topological space X into S(n), and for all x the values f(x) and g(x) are not antipodal, then f and g are homotopic. In particular, a nonsurjective map into S(n) is nullhomotopic.

(iii) A map f from S(n) into some topological space Y is nullhomotopic if and only if f has a continuous extension to the (n+1)-ball (points of norm at most 1 in (n+1)-space).

Relative homotopy

Let X,Y be two topological spaces, and A a subspace of X. A homotopy F from X × I to Y is called homotopy relative to A if for each a in A the map F(a,t) is constant (independent of t).

3.2 Homotopy classes

Being homotopic is an equivalence relation, so we have equivalence classes. Given two spaces X,Y, and a map f from X to Y, let [f] denote the homotopy class of f, that is, the set of all maps from X to Y homotopic to f. Let [X,Y] denote the set of homotopy classes of maps from X to Y.

Under suitable assumptions homotopy classes are precisely the path-components in the space C(X,Y) of continuous functions from X to Y. [E.g., give C(X,Y) the compact-open topology, and assume that X is a k-space, i.e., has the topology defined by the injection maps from its compact subspaces. Then this statement holds.]

Again, we can look at relative homotopy, and similar things hold.

Exercise Let f be a path (a map from the unit interval I into some topological space X). Let p be a continuous map from I to itself fixing 0 and 1 (a parameter transformation). Show that [f] = [fp] (where juxtaposition denotes composition: (fp)(t) = f(p(t))).

3.3 Fundamental group

We call two paths f,g (maps from I into some space X) homotopic when they are homotopic relative to {0,1}. Given a path f, the inverse path is the path f' defined by f'(t) = f(1-t). Given two paths f,g, where the endpoint f(1) of the first is the starting point g(0) of the second, their product is the path h = f#g defined by h(t) = f(2t) for 2t < 1 and h(t) = g(2t-1) otherwise.

Theorem Let X be a topological space and fix a point x in it. Let e(x) be the constant map that sends all of I to x. Let P(X;x) be the set of homotopy classes of paths starting and ending in x. Then P(X;x) is a group with respect to the multiplication defined by [f][g] = [f#g]. It has inverses defined by [f]' = [f'] and unit element [e(x)]. When X is path-connected, all groups P(X;x) are isomorphic.

Proof We have to check that the operations are well-defined. They are.

The group found here is called the Poincaré group or fundamental group of X (at the point x).

Exercise (i) Show that the fundamental group of the circle S(1) is Z, the additive group of the integers. (ii) Show that the fundamental group of the sphere S(n) with n > 1 is trivial.

3.4 Morphisms

Theorem Let X,Y be two spaces, and let f be a map from X to Y. We find a homomorphism f* from P(X;x) to P(Y;f(x)) by sending a path p to the path fp (defined by fp(t) = f(p(t))).

This is a functor from pointed topological spaces with maps to groups with homomorphisms. In particular: (gf)* = g* f* and the identity map on X is sent to the identity homomorphism on P(X;x).

It follows that if f is a homotopy equivalence from X to Y (that is, a map such that there is a map g from Y to X such that both fg and gf are nullhomotopic), then f* is an isomorphism.

3.5 Simple connectedness

A topological space X is called simply connected when it is path-connected and its fundamental group P(X) is trivial.

Exercise A path-connected space X is simply connected if and only if each map from the circle S(1) into X can be extended to a map from the 2-ball (circle plus interior) into X.

3.6 The fundamental group of a product

Theorem The fundamental group of a product is the product of the fundamental groups of the factors.

Proof If X is a product of factors Xi, and pi is the projection onto Xi, then pi* maps P(X;x) into P(Xi;xi), and the homomorphism p with i-th coordinate pi maps P(X;x) into the product of the P(Xi;xi). Now p is trivially surjective and injective, so is an isomorphism.

Exercise Show that the product of simply connected spaces is simply connected again.

3.7 The fundamental group of a retract

Let X be a topological space. A subspace A of X is called a retract of X if there is a map r from X onto A that extends the identity on A. The subspace A is called a deformation retract of X if there is a map r from X onto A that is homotopic to the identity on X relative to A.

Exercise Clearly, every deformation retract is a retract. Show that the converse does not hold.

Given a topological space X and a subspace A, let i be the inclusion map. Let a be a point of A. Then i* is a homomorphism from P(A;a) into P(X;a).

Exercise Show that i* need not be injective.

Theorem If A is a retract of X, then i* is injective.

Theorem If A is a deformation retract of X, then i* is an isomorphism.

Exercise Show that a retract of a simply connected space is again simply connected.

3.8 The fundamental group of a union

Let X be a topological space that is the union of two path-connected subspaces A and B, where the intersection of A and B is nonempty and path-connected. Then the fundamental group of X is generated by (the images of) the fundamental groups of A and B.

More generally we have: Theorem (Van Kampen) Under these same hypotheses, let f and g be homomorphisms from P(A;c) (resp. P(B;c)) into some group G that agree on the intersection C of A and B. (Here c is a point of C.) Then there is a unique homomorphism h from P(X;c) into G that agrees with f and g on A and B, respectively.

Perhaps more hypotheses are required?

3.9 The fundamental group of a topological group is Abelian

If X is a topological group then we have a multiplication of paths given by (f.g)(t) = f(t).g(t) where . denotes the group operation.

Theorem The group operation on X induces a group operation on P(X;x) that coincides with the old group operation, and P(X;x) is commutative.

Proof First of all [f].[g] = [f.g] is well-defined. Next, if e = e(x), then [f].[g] = [f#e].[e#g] = [f#g] = [f][g] and [f].[g] = [e#f].[g#e] = [g#f] = [g][f].

3.10 The fundamental group of orthogonal groups

The fundamental group of SO(2,R), the group of orthogonal transformations of determinant 1 of the real plane, is isomorphic to Z, the additive group of the integers. (Indeed, this group is isomorphic to S(1).) The fundamental group of SO(n,R) with n > 2 is isomorphic to Z2, the additive group of the integers mod 2.

3.11 Hopf spaces

A Hopf space is a space in which the proof given above for the statement that the fundamental group of a topological group is Abelian still works. Thus, by definition, the fundamental group of a Hopf space is Abelian. What do we need? A multiplication . on the space that preserves homotopy so that it induces a multiplication on homotopy classes, and a unit element e for this multiplication such that [e#f] = [f] = [f#e].

3.12 Higher homotopy groups

Instead of using maps from I into X, with the condition that both endpoints are mapped to a given point, we consider maps from I^n into X, with the condition that the entire boundary (the set of all points that have at least one coordinate 0 or 1) is mapped to a given fixed point x. Now again we have a composition # (acting on the first coordinate only) that induces a group operation on the homotopy classes. The group obtained is called pi_n(X;x). (Thus, pi_1(X;x) = P(X;x).)

Let L(X;x) be the space of loops in X with base point x (i.e., maps from I to X sending 0 and 1 to x), provided with the compact-open topology. For suitably nice (say, locally compact) X we have pi_2(X;x) = P(L(X;x);e) when e is the constant map that sends I to x, and more generally pi_n(X;x) = P(L_(n-1)(X;x);e) where L_n(X;x) = L(L_(n-1)(X;x);e) for a suitable constant e = e_n.

For n > 1 the group pi_n(X;x) is Abelian. Indeed, L(X;x) is a Hopf space for the operation #.

3.13 Fiber spaces

Let E and B be topological spaces (the total and base space, respectively), and let p : E -> B be a map (the projection). We say that p has the CHP (Covering Homotopy Property) for X if for every map F : X×I -> B and every map f' : X -> E with F(x,0) = pf'(x) (for all x) there is a map F' : X×I -> E with F = pF' and F'(x,0) = f'(x) (for all x). In this case (E,B,p) is also called a fiber space (for X).

We say that p has the BP (Bundle Property) if there is a space D such that B has an open cover such that for each member U of this cover there is a homeomorphism f_U : U × D -> p^(-1) (U) with p f_U (u,d) = u (for all u,d). In this case (B,E,p) is also called locally trivial.

Theorem If p has the Bundle Property, then it has the CHP for every paracompact Hausdorff space X. If moreover B is paracompact, then p has the CHP for every topological space.

For a proof, see e.g. Dugundji, Topology, Allyn & Bacon, 1966, Chapter XX.

If (S(n),B,p) is a fiber space, and B has more than one point, then the map p is not nullhomotopic. Indeed, otherwise the homotopy could be lifted to one from the identity to a map with image contained in a single fiber, so that the identity on S(n) would be nullhomotopic. But it isn't, for example because S(n) has nontrivial homology.

Hopf gave as examples locally trivial fiber structures S(3) -> S(2) and S(7) -> S(4) and S(15)->S(8) constructed from the norm 1 elements in the complex, quaternion or octonion plane, mapped to the corresponding point of the projective line.

These maps are algebraically trivial, that is, they induce 0 on the homology and cohomology groups, but homotopically nontrivial. Thus, homotopy is strictly finer than homology.

3.14 Some results for spheres

A table with some results for spheres, taken from Sze-Tsen Hu, Homotopy Theory, Academic Press, 1959.

Let S(n) be the n-sphere, and consider P(m,n) = pi_m(S(n)) for n > 0. If m < n then this group is trivial. If m = n then it is isomorphic to Z, the additive group of the integers. If n = 1 and m > n, then it is trivial. We have P(m,2) = P(m,3) for every m > 2. If n is odd and m > n, then P(m,n) is finite.

If n > 2 and m = n+1 or m = n+2 then it is cyclic of order 2.

P(6,3) = Z(12), the cyclic group of order 12. P(7,4) = Z + Z(12). If n > 4 and m = n+3 then P(m,n) = Z(24).

P(7,3) = Z(2), P(8,4) = Z(2)+Z(2), P(9,5) = Z(2). If n > 5 and m = n+4 then P(m,n) = 0.

P(8,3) = Z(2), P(9,4) = Z(2)+Z(2), P(10,5) = Z(2), P(11,6) = Z. If n > 6 and m = n+5 then P(m,n) = 0.

P(9,3) = Z(3), P(10,4) = Z(24)+Z(2). If n > 4 and m = n+6 then P(m,n) = Z(2).

P(10,3) = P(11,4) = Z(15), P(12,5) = Z(30), P(13,6) = Z(60), P(14,7) = Z(120), P(15,8) = Z+Z(120). If n > 8 and m = n+7 then P(m,n) = Z(240).

P(11,3) = P(12,4) = P(13,5) = Z(2), P(14,6) = Z(24) + Z(2), P(15,7) = Z(2)+Z(2)+Z(2), P(16,8) = Z(2)+Z(2)+Z(2)+Z(2), P(17,9) = Z(2)+Z(2)+Z(2). If n > 9 and m = n+8 then P(m,n) = Z(2)+Z(2).

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