Self-maps of the product of two spheres fixing the diagonal

AbstractWe compute the monoid of essential self-maps of Sn×Sn fixing the diagonal. More generally, we consider products S×S, where S is a suspension. Essential self-maps of S×S demonstrate the interplay between the pinching action for a mapping cone and the fundamental action on homotopy classes under a space. We compute examples with non-trivial fundamental actions

The third homotopy group as a π₁-module

It is well-known how to compute the structure of the second homotopy group of a space, X, as a module over the fundamental group π₁X, using the homology of the universal cover and the Hurewicz isomorphism. We describe a new method to compute the third homotopy group, π₃X as a module over π₁X. Moreover, we determine π₃X as an extension of π₁X-modules derived from Whitehead's Certain Exact Sequence. Our method is based on the theory of quadratic modules. Explicit computations are carried out for pseudo-projective 3-spaces X=S¹Ue²Ue³ consisting of exactly one cell in each dimension ≤ 3

Presentation of homotopy types under a space

We compare the structure of a mapping cone in the category Top^D of spaces under a space D with differentials in algebraic models like crossed complexes and quadratic complexes. Several subcategories of Top^D are identified with algebraic categories. As an application we show that there are exactly 16 essential self--maps of S^2 x S^2 fixing the diagonal.Comment: 21 page

Mary Lena Bleile, Cello

Sonata No. 5 in D major / Ludwig van Beethoven; Sonata No. 2 in D major / Johann Sebastian Bach; Fratres / Arvo Pär

On the sensitivity of the solitary wave profile recovery formula to wave speed noise

We derive a bound on the error in the recovery of the profile of an irrotational solitary water wave from pressure data given noise in the measurement of the wave speed. First, we prove that Constantin's exact solitary wave reconstruction formula is well-defined in the sense of functions given that the wave speed error is sufficiently small. We then analytically prove that the error in the reconstruction is bounded and obtain a formula for this bound. Finally, we compare the estimate with elementary numerical experiments