M8130 Algebraic topology, tutorial 12, 2020 13. 1. 2021
Exercise 1. Show that (n−1)-connected compact manifold of dim n is homotopy equivalent
to Sn
(n ≥ 2).
Solution. Take manifold M as two parts: disk and its complement. Map the disk to
Sn
= Dn
/Sn−1
, that is the identity on the interior of the disk and the complement goes
to one point. We have f : M → Sn
and for i < n by Hurewicz theorem (M is (n − 1)connected)
we get an iso f∗ : Hi(M) ∼= Hi(Sn
) = 0. The same iso we obtain for i > n
(both Hi are zero).
We conclude our application with this spectacular diagram where all arrows are iso,
reasoning goes from the bottom arrows by excision, then with deniton of fundamental
class we get the top arrow (our main focus) an iso. (we denote interior of B as Bo
)
Hn(M) //

Hn(Sn
)

Hn(M, M − (Bn
)o
) //

Hn(Sn
, Sn
− (Bn
)o
)

Hn(disk, ∂disk) id
// Hn(disk, ∂disk)
Remark. We have got that a compact manifold 3-dim (compact) which is 2-connected is
homotopy equivalent to S3
. There is another result: Every 3-dim manifold simply connected
compact manifold is homeomorphic to S3
. This latter result (it might seem we are close
to proving it) is actually famous Poincaré conjecture, one of Millenium Prize Problems
and it was already solved by Grigori Perelman in 2002. Interesting story and interesting
mathematician for sure. Perelman declined Fields medal (among other prizes).