Talk:Normal bundles in products of spheres (Ex)
1. The normal bundle of the diagonal embedding is isomorphic to the tangent bundle of
. The tangent bundle of
is trivial if and only if
, by Adam's theorem.
2. We present two solutions.
Low-tech solution: Given immersions and
of closed manifolds, one can produce an immersion
by "connecting the immersions by a tube". (See for instance the definition of the sum in
in Lück's book.) If both immersions happen to have a trivial normal bundle, then the immersion
also has a trivial normal bundle.
We use this for and
, where the first immersion is the inclusion
and the second one is the corresponding inclusion
into the second factor. Both these embeddings apparently have trivial normal bundles. Hence so does their "sum"
![\displaystyle S^k \cong S^k \# S^k \to S^k\times S^k.](/images/math/7/2/7/727085118e1f7084e2cbc1fae1699da2.png)
The homotopy class of this map is given by where the first map is the pinch map. By definition of the sum in
, it is thus the sum
. Its Hurewicz image is given by the diagonal class
.
High-tech solution: The Hirsch-Smale classification of immersions implies that
![\displaystyle \pi_0 \mathrm{Imm}(S^k, S^k\times S^k) \cong \mathrm{colim}_n \; \pi_0 \mathrm{Mono}(TS^k\oplus \varepsilon^n, TS^k \times TS^k\oplus \varepsilon^n)](/images/math/a/8/7/a871a820dfe5c6597fa4db6c1158dabe.png)
where denotes the
-dimensional trivial bundle.
We will use surjectivity. Both the bundles and
over
are trivial, thus isomorphic to
and
respectively. However the inclusion
of the first
summands is a bundle monomorphism with trivial complement.
By Smale-Hirsch, the homotopy class of the composite
![\displaystyle \xymatrix{ TS^k \oplus \varepsilon \ar[r] \ar[d] & TS^k \oplus TS^k \oplus \varepsilon \ar[r]\ar[d] & (TS^k \times TS^k) \oplus \varepsilon \ar[d]\\ S^k \ar[r]^{\mathrm{id}} & S^k \ar[r]^{\Delta} & S^k\times S^k }](/images/math/2/0/2/2028683a826ea5b74f0ce403a345800d.png)
is the differential of an immersion.
Its normal bundle is still trivial. Moreover, by construction, this immersion is (non-regularly) homotopic to the diagonal. So its Hurewicz image is the diagonal class .