The Kervaire invariant in homotopy theory
نویسنده
چکیده
In this note we discuss how the first author came upon the Kervaire invariant question while analyzing the image of the J-homomorphism in the EHP sequence. One of the central projects of algebraic topology is to calculate the homotopy classes of maps between two finite CW complexes. Even in the case of spheres – the smallest non-trivial CW complexes – this project has a long and rich history. Let S denote the n-sphere. If k < n, then all continuous maps S → S are null-homotopic, and if k = n, the homotopy class of a map S → S is detected by its degree. Even these basic facts require relatively deep results: if k = n = 1, we need covering space theory, and if n > 1, we need the Hurewicz theorem, which says that the first non-trivial homotopy group of a simply-connected space is isomorphic to the first non-vanishing homology group of positive degree. The classical proof of the Hurewicz theorem as found in, for example, [28] is quite delicate; more conceptual proofs use the Serre spectral sequence. Let us write πiS for the ith homotopy group of the n-sphere; we may also write πk+nS to emphasize that the complexity of the problem grows with k. Thus we have πn+kS = 0 if k < 0 and πnS ∼= Z. Given the Hurewicz theorem and knowledge of the homology of Eilenberg-MacLane spaces it is relatively simple to compute that πn+1S n ∼= 0, n = 1; Z, n = 2; Z/2Z, n ≥ 3. The generator of π3S is the Hopf map; the generator in πn+1S, n > 2 is the suspension of the Hopf map. If X has a basepoint y, the suspension ΣX is given by ΣX = S ×X/(S × y ∪ 1×X) where 1 ∈ S ⊆ C. Then ΣS ∼= S and we get a suspension homomorphism E : πn+kS → π(n+1)+kS. ∗The second author was partially supported by the National Science Foundation.
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