# Algebraic topology - Errata by Allen Hatcher By Allen Hatcher

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5, the conclusion follows. 3. This follows by a similar argument. The hypothesis on the manifold guarantees that every map into a sphere of the same dimension is null-homotopic. § 4. Which homotopy spheres bound parallelizable manifolds? Deﬁne a subgroup bPn+1 ⊂ Θn as follows. A homotopy n-sphere M represents an element of bPn+1 if and only if M is a boundary of a parallelizable manifold. We will see that this condition depends only on the August 26, 2009 16:21 18 9in x 6in b789-ch02 M. Kervaire and J.

8. M. Kervaire. Relative characteristic classes, Amer. J. Math. 79 (1957), 517–558. 9. M. Kervaire. An interpretation of G. Whitehead’s generalization of the Hopf invariant, Ann. Math. 69 (1959), 345–364. 10. M. Kervaire. A note on obstructions and characteristic classes, Amer. J. Math. 81 (1959), 773–784. 11. M. Kervaire. Some non-stable homotopy groups of Lie groups, Illinois J. Math. 4 (1960), 161–169. 12. M. Kervaire. A manifold which does not admit any diﬀerentiate structure, Comment. Math.

Thus j(α) can be chosen so that −l < l − lj(α) ≤ l. This choice of j(α) will guarantee an improvement except in the special case where l happens to be divisible by l. Our progress so far can be summarized as follows. 3. Let M be a framed (k − 1)-connected manifold of dimension 2k + 1 with odd k, k > 1, such that Hk M is ﬁnite. Let χ(ϕ, F ) be a framed modiﬁcation of M which replaces the element λ ∈ Hk M of order l > 1 by an element λ ∈ Hk M of order ±l . If l ≡ 0 mod l, then it is possible to choose (α) ∈ πk (SOk+1 ) so that the modiﬁcation χ(ϕ) can still be framed, and so that the group Hk Mα is deﬁnitely smaller than Hk M .