28 COVERING SPACES AND THE FUNDAMENTAL GROUP 209

3. Deduce from (16.27.10) that the Hopf fibration of S3 over S2 (16.14.10) is not trivializ-
able.

4. Let X be a simply-connected, arcwise-connected space. If a, b are any two points of X,
show that the set Efl, b consists of a single element. (If y, y e jQfl, b, show that y is
equivalent (with respect to R«, /,) to (yy°)y'.)

5. Let X be a locally compact metrizable space, and let A, B be closed subsets of X such
that X = A u B. Suppose that A, B are each arcwise-connected and simply-connected;
suppose also that A n B is arcwise-connected and that, for each x e A n B, there exists
a fundamental system of open neighborhoods V of x for each of which V n B is arcwise-
connected and simply-connected, and V n A n B arcwise-connected. Show that X is
arcwise-connected and simply-connected. (Let y : I->X be a loop with origin ae A,
and suppose that y(I) meets B. Then the inverse image under y of y(I) n (X — A) is the
union of a (finite or infinite) sequence of pairwise disjoint open intervals Im in I. For
each integer nt cover y(I) by a finite number of open subsets U«, „ of X, of diameter
^ 1//Z, such that U«, „ n B is arcwise-connected and simply-connected, and U«, „ n A n B
arcwise-connected. Then consider the finite set of intervals I* such that y(I*) is con-
tained in U«, „ but not in Ua> n+i; finally, use Problem 4.) Hence give another proof of
the fact that Sn is simply connected for n J> 2.

28. COVERING SPACES AND THE FUNDAMENTAL GROUP

(16.28.1) Let (X, B9jp) be a covering of a differential manifold B (16.12.4),
fa continuous mapping of a connected topological space Z into B, and glt
g2: Z-+X two continuous liftings (16.12.1) off. If there exists aeZ such that

The set of points z e Z such that #i(z) = g2(z) is nonempty and closed in
Z (12.3.5); hence it is enough to show that it is also open in Z. Now, if z0 e Z
is such that gi(z0) = g2(z0) == xo > there exists an open neighborhood U of ;c0
such that p\U is a homeomorphism of U onto the open set p(V) c B. Since
gl9 g2 are continuous, there exists a neighborhood V of z0 in Z such that
^(V) c U and gffl) c U. However, for all z e V, we have X0i(z)) =/(X) —
p(g2(z))l since p \ U is injective, it follows that g^z) = g2(z) for all z e V. This
completes the proof.

(16.28.2) Let (X, B, p) be a covering of a differential manifold B.

(i) Fpr each path p : I -* B with origin b e B, and for each point a ep~l(b)9
there exists a unique path y : I -> X with origin a such that ft = p o y (called the
lifting of P, with origin a).rphic to an