On the pontryagin-steenrod-wu theorem

On the Pontryagin-steenrod-wu Theorem

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This paper is on the homotopy classification of maps of (n+1)-dimensional manifolds into the n-dimensional sphere. For a continuous map f : M → S define the degree deg f ∈ H1(M;Z) to be the class dual to f[S], where [S] ∈ H(S;Z) is the fundamental class. We present a short and direct proof of the following specific case of the Pontryagin-Steenrod-Wu theorem: Theorem. Let M be a connected orient...

A Note on the Hayman-wu Theorem

The Hayman-Wu theorem states that the preimage of a line or circle L under a conformal mapping from the unit disc D to a simplyconnected domain Ω has total Euclidean length bounded by an absolute constant. The best possible constant is known to lie in the interval [π, 4π), thanks to work of Øyma and Rohde. Earlier, Brown Flinn showed that the total length is at most π in the special case in whi...

Bordism and the Pontryagin-Thom Theorem

Given the classification of low dimensional manifolds up to equivalence relations such as diffeomorphism or homeomorphism, one would hope to be able to continue to classify higher dimensional manifolds. Unfortunately, this turns out to be difficult or impossible, and so one solution would be turn to some weaker equivalence relation. One such equivalence relation would be to consider manifolds u...

On the secondary Steenrod algebra

We introduce a new model for the secondary Steenrod algebra at the prime 2 which is both smaller and more accessible than the original construction of H.-J. Baues. We also explain how BP can be used to define a variant of the secondary Steenrod algebra at odd primes.