By Milgram R.J.

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1 2 3 4 .. .. ... ... ... ... ... . ... ... .. ... ... .. ... ..... ..... ..... . . ..... ..... ...... ...... ....... ........ . . . ......... ............ ............................................................. • v1 α • v2 α • v3 α • v4 α • v5 The example for n = 5 The fundamental group of this manifold is the free group on (n−1)-generators corresponding to the arcs in the positive sense around the (n − 1) deleted disks.

This can be seen by using the explicit maps of Rn × Dn to the tangent bundles at the respective hemispheres by using the rotation Rθ,x which rotates through the angle θ in the subspace of Rn+1 spanned by (x, 0) and (0, 1) and is the identity in the perpendicular complement. On the other hand, consider the map f : Rn+1 − →R2n defined by (t, v) → cos(2πt)v . −sin(2πt)v 40 3. IMMERSIONS AND EMBEDDINGS The differential df at the point (t, x) is given as the 2n × n + 1 matrix −2πsin(2πt)x cos(2πt)In .

Since m ≥ 5 g is homotopic to an embedding leaving f fixed. Now we ensure that V ∩g(D2 ) = ∅. By general position we can move g(D2 ) away from V by an arbitrarily small perturbation leaving g an embedding, and leaving f alone on S 1 . The result is an embedded g(D2 ) ⊂ M \V with ∂(g(D2 )) = f (S 1 ), so that x = 1 ∈ π1 (M \V ). Returning to the proof of 14, apply 12 to obtain a differentiably embedded disk D2 ⊂ M \f (N ) with ∂(D2 ) = γ˜ and boundary one of the two boundary components of the embedded D1 × S 1 above.