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Heaps and unpointed stable homotopy theory

Lukáš Vokřínek — 2014

Archivum Mathematicum

In this paper, we show how certain “stability phenomena” in unpointed model categories provide the sets of homotopy classes with a canonical structure of an abelian heap, i.e. an abelian group without a choice of a zero. In contrast with the classical situation of stable (pointed) model categories, these sets can be empty.

Computing the abelian heap of unpointed stable homotopy classes of maps

Lukáš Vokřínek — 2013

Archivum Mathematicum

An algorithmic computation of the set of unpointed stable homotopy classes of equivariant fibrewise maps was described in a recent paper [4] of the author and his collaborators. In the present paper, we describe a simplification of this computation that uses an abelian heap structure on this set that was observed in another paper [5] of the author. A heap is essentially a group without a choice of its neutral element; in addition, we allow it to be empty.

A generalization of Thom’s transversality theorem

Lukáš Vokřínek — 2008

Archivum Mathematicum

We prove a generalization of Thom’s transversality theorem. It gives conditions under which the jet map f * | Y : Y J r ( D , M ) J r ( D , N ) is generically (for f : M N ) transverse to a submanifold Z J r ( D , N ) . We apply this to study transversality properties of a restriction of a fixed map g : M P to the preimage ( j s f ) - 1 ( A ) of a submanifold A J s ( M , N ) in terms of transversality properties of the original map f . Our main result is that for a reasonable class of submanifolds A and a generic map f the restriction g | ( j s f ) - 1 ( A ) is also generic. We also present an example of A where the...

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