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University of the Basque Country, Euskal Herriko Unibertsitatea
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Papers by Peter Eccles
An Introduction to Mathematical Reasoning, Dec 11, 1997
Definition 0.0.1 Fix n ∈ Z with n > 1. We say that two integers a and b are congruent modulo n an... more Definition 0.0.1 Fix n ∈ Z with n > 1. We say that two integers a and b are congruent modulo n and write a ≡ b (mod n) provided that n | (b − a).
Let a and b be integers with b ≠ 0. Then there exists integers q and r such that: a = bq + r and ... more Let a and b be integers with b ≠ 0. Then there exists integers q and r such that: a = bq + r and 0 ≤ r ≤ | b | In fact the integers q and r are unique. The rule that b must be non-negative is convention and guarantees uniqueness.
We consider the problem of detecting Mahowald's family ηi ∈ 2π S 2 i in homology. This allows us ... more We consider the problem of detecting Mahowald's family ηi ∈ 2π S 2 i in homology. This allows us to identify specific spherical classes in H * Ω 2 i+1 −8+k 0 S 2 i −2 for 0 k 6. We then identify the type of the subalgebras that these classes give rise to, and calculate the A-module and R-module structure of these subalgebras. We shall the discuss the relation of these calculations to the Curtis conjecture on spherical classes in H * Q0S 0 .
Cambridge University Press, Dec 1, 1997
An Introduction to Mathematical Reasoning
Proceedings of the Steklov Institute of Mathematics, 2006
We present a geometrical version of Herbert's Theorem determining the homology classes represente... more We present a geometrical version of Herbert's Theorem determining the homology classes represented by the multiple point manifolds of a self-tranverse immersion. Herbert's Theorem and generalizations can readily be read off from this result. The simple geometrical proof is based on ideas in Herbert's paper. We also describe the relationship between this theorem and the homotopy theory of Thom spaces.
We consider the problem of calculating the Hurewicz image of Mahowald's family η_i∈_2π_2^i^S.... more We consider the problem of calculating the Hurewicz image of Mahowald's family η_i∈_2π_2^i^S. This allows us to identify specific spherical classes in H_*Ω_0^2^i+1-8+kS^2^i-2 for 0≤ k≤ 6. We then identify the type of the subalgebras that these classes give rise to, and calculate the A-module and R-module structure of these subalgebras. We shall the discuss the relation of these calculations to the Curtis conjecture on spherical classes in H_*Q_0S^0, and relations with spherical classes in H_*Q_0S^-n.
A well-known formula of R.J. Herbert’s relates the various homology classes represented by the se... more A well-known formula of R.J. Herbert’s relates the various homology classes represented by the self-intersection immersions of a self-transverse immersion. We prove a geometrical version of Herbert’s formula by considering the self-intersection immersions of a self-transverse immersion up to bordism. This clarifies the geometry lying behind Herbert’s formula and leads to a homotopy commutative diagram of Thom complexes. It enables us to generalise the for-mula to other homology theories. The proof is based on Herbert’s but uses the relationship between self-intersections and stable Hopf invariants and the fact that bordism of immersions gives a functor on the category of smooth manifolds and proper immersions.
An Introduction to Mathematical Reasoning, Dec 11, 1997
Definition 0.0.1 Fix n ∈ Z with n > 1. We say that two integers a and b are congruent modulo n an... more Definition 0.0.1 Fix n ∈ Z with n > 1. We say that two integers a and b are congruent modulo n and write a ≡ b (mod n) provided that n | (b − a).
Let a and b be integers with b ≠ 0. Then there exists integers q and r such that: a = bq + r and ... more Let a and b be integers with b ≠ 0. Then there exists integers q and r such that: a = bq + r and 0 ≤ r ≤ | b | In fact the integers q and r are unique. The rule that b must be non-negative is convention and guarantees uniqueness.
We consider the problem of detecting Mahowald's family ηi ∈ 2π S 2 i in homology. This allows us ... more We consider the problem of detecting Mahowald's family ηi ∈ 2π S 2 i in homology. This allows us to identify specific spherical classes in H * Ω 2 i+1 −8+k 0 S 2 i −2 for 0 k 6. We then identify the type of the subalgebras that these classes give rise to, and calculate the A-module and R-module structure of these subalgebras. We shall the discuss the relation of these calculations to the Curtis conjecture on spherical classes in H * Q0S 0 .
Cambridge University Press, Dec 1, 1997
An Introduction to Mathematical Reasoning
Proceedings of the Steklov Institute of Mathematics, 2006
We present a geometrical version of Herbert's Theorem determining the homology classes represente... more We present a geometrical version of Herbert's Theorem determining the homology classes represented by the multiple point manifolds of a self-tranverse immersion. Herbert's Theorem and generalizations can readily be read off from this result. The simple geometrical proof is based on ideas in Herbert's paper. We also describe the relationship between this theorem and the homotopy theory of Thom spaces.
We consider the problem of calculating the Hurewicz image of Mahowald's family η_i∈_2π_2^i^S.... more We consider the problem of calculating the Hurewicz image of Mahowald's family η_i∈_2π_2^i^S. This allows us to identify specific spherical classes in H_*Ω_0^2^i+1-8+kS^2^i-2 for 0≤ k≤ 6. We then identify the type of the subalgebras that these classes give rise to, and calculate the A-module and R-module structure of these subalgebras. We shall the discuss the relation of these calculations to the Curtis conjecture on spherical classes in H_*Q_0S^0, and relations with spherical classes in H_*Q_0S^-n.
A well-known formula of R.J. Herbert’s relates the various homology classes represented by the se... more A well-known formula of R.J. Herbert’s relates the various homology classes represented by the self-intersection immersions of a self-transverse immersion. We prove a geometrical version of Herbert’s formula by considering the self-intersection immersions of a self-transverse immersion up to bordism. This clarifies the geometry lying behind Herbert’s formula and leads to a homotopy commutative diagram of Thom complexes. It enables us to generalise the for-mula to other homology theories. The proof is based on Herbert’s but uses the relationship between self-intersections and stable Hopf invariants and the fact that bordism of immersions gives a functor on the category of smooth manifolds and proper immersions.