Normal bundles in products of spheres (Ex)

Exercise 0.1.

1. For which $k$$<wikitex>; {{beginthm|Exercise}} # For which $k$ does the diagonal embedding $S^k \to S^k \times S^k, \quad x \mapsto (x, x)$ have trivial normal bundle? # For all $k$, find an immersion representing the diagonal homology class $(1, 1) \in H_k(S^k \times S^k)$ with trivial normal bundle. {{endthm}} {{beginrem|Hint}} You may wish to look at [[Immersing n-spheres in 2n-space (Ex)]] for part 2. {{endrem}} </wikitex> == References == {{#RefList:}} [[Category:Exercises]] [[Category:Exercises with solution]]k$ does the diagonal embedding $S^k \to S^k \times S^k, \quad x \mapsto (x, x)$$S^k \to S^k \times S^k, \quad x \mapsto (x, x)$ have trivial normal bundle?
2. For all $k$$k$, find an immersion representing the diagonal homology class $(1, 1) \in H_k(S^k \times S^k)$$(1, 1) \in H_k(S^k \times S^k)$ with trivial normal bundle.

Hint 0.2. You may wish to look at Immersing n-spheres in 2n-space (Ex) for part 2.