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58
Algebraic topology and modular forms
 Proceedings of the International Congress of Mathematicians, Vol. I (Beijing, 2002), Higher Ed
, 2002
"... The problem of describing the homotopy groups of spheres has been fundamental to algebraic topology for around 80 years. There were periods when specific computations were important and periods when the emphasis favored theory. Many ..."
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Cited by 54 (3 self)
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The problem of describing the homotopy groups of spheres has been fundamental to algebraic topology for around 80 years. There were periods when specific computations were important and periods when the emphasis favored theory. Many
On the nonexistence of elements of Kervaire invariant one
, 2009
"... We show that the Kervaire invariant one elements θj ∈ π 2 j+2 −2 S 0 exist only for j ≤ 6. By Browder’s Theorem, this means that smooth framed manifolds of Kervaire invariant one exist only in dimensions 2, 6, 14, 30, 62, and possibly 126. Except for dimension 126 this resolves a longstanding prob ..."
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Cited by 46 (7 self)
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We show that the Kervaire invariant one elements θj ∈ π 2 j+2 −2 S 0 exist only for j ≤ 6. By Browder’s Theorem, this means that smooth framed manifolds of Kervaire invariant one exist only in dimensions 2, 6, 14, 30, 62, and possibly 126. Except for dimension 126 this resolves a longstanding problem in algebraic topology.
Surgery and the generalized Kervaire invariant
 Universitdt Bielefeld
, 1985
"... framed manifolds of dimension 4k+ 2 (see [12]) was an important stimulant for the development of surgery theory; but it also led to the theory of the 'generalized Kervaire Invariant ' of Browder and Brown [2, 3]. ..."
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Cited by 13 (1 self)
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framed manifolds of dimension 4k+ 2 (see [12]) was an important stimulant for the development of surgery theory; but it also led to the theory of the 'generalized Kervaire Invariant ' of Browder and Brown [2, 3].
Betaelements and divided congruences
 Amer. J. Math
"... The finvariant is an injective homomorphism from the 2line of the AdamsNovikov spectral sequence to a group which is closely related to divided congruences of elliptic modular forms. We compute the finvariant for two infinite families of βelements and explain the relation of the arithmetic of d ..."
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Cited by 10 (2 self)
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The finvariant is an injective homomorphism from the 2line of the AdamsNovikov spectral sequence to a group which is closely related to divided congruences of elliptic modular forms. We compute the finvariant for two infinite families of βelements and explain the relation of the arithmetic of divided congruences with the Kervaire invariant one problem. 1.
Almost Nonnegative Curvature and Cohomogeneity One
, 2001
"... We show that any closed cohomogeneity one manifold supports metrics of almost nonnegative sectional curvature which are moreover invariant under the cohomogeneity one action, thereby establishing a conjecture of Grove and Ziller in the almost nonnegatively curved setting. Applications of our result ..."
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Cited by 9 (3 self)
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We show that any closed cohomogeneity one manifold supports metrics of almost nonnegative sectional curvature which are moreover invariant under the cohomogeneity one action, thereby establishing a conjecture of Grove and Ziller in the almost nonnegatively curved setting. Applications of our result include that there are infinitely many dimensions in which there exist almost nonnegatively curved topological spheres and homotopy lens as well as homotopy real projective spaces, all differentiably distinct from the standard ones.
Spherical classes and the algebraic transfer
 Trans. Amer. Math. Soc
"... Abstract. We study a weak form of the classical conjecture which predicts that there are no spherical classes in Q0S0 except the elements of Hopf invariant one and those of Kervaire invariant one. The weak conjecture is obtained by restricting the Hurewicz homomorphism to the homotopy classes which ..."
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Abstract. We study a weak form of the classical conjecture which predicts that there are no spherical classes in Q0S0 except the elements of Hopf invariant one and those of Kervaire invariant one. The weak conjecture is obtained by restricting the Hurewicz homomorphism to the homotopy classes which are detected by the algebraic transfer. Let Pk = F2[x1; : : : ; xk] with jxij = 1. The general linear group GLk = GL(k;F2) and the (mod 2) Steenrod algebra A act on Pk in the usual manner. We prove that the weak conjecture is equivalent to the following one: The canonical homomorphism jk: F2 A