arXiv subjects

Arley Fernando Torres

Publications and source records attributed to Arley Fernando Torres.

2 recordsLinked to original sources

arXiv2023

Thom's counterexamples for the Steenrod problem

The present paper deals with integral classes $ξ_p\in H_{2p+1}(L^{2p+1}\times L^{2p+1})$ which are counterexamples for the Steenrod realization problem, where $L^{2p+1}$ is the $(2p+1)$-dimensional lens space and $p\geq 3$ is a prime number. For $p=3$, this is Thom's famous counterexample. We give a geometric description of this class using the theory of stratifolds. As a consequence, we obtain a geometric interpretation of the obstruction to realizability in terms of the Atiyah--Hirzebruch spectral sequence.

math.AT↗

arXiv2023

$\mathbb{Z}_k$-stratifolds

Generalizing the ideas of $\mathbb{Z}_k$-manifolds from Sullivan and stratifolds from Kreck, we define $\mathbb{Z}_k$-stratifolds. We show that the bordism theory of $\mathbb{Z}_k$-stratifolds is sufficient to represent all homology classes of a $CW$-complex with coefficients in $\mathbb{Z}_k$. We present a geometric interpretation of the Bockstein long exact sequences and the Atiyah-Hirzebruch spectral sequence for $\mathbb{Z}_k$-bordism ($k$ an odd number). Finally, for $p$ an odd prime, we give geometric representatives of all classes in $H_*(B\mathbb{Z}_p;\mathbb{Z}_p)$ using $\mathbb{Z}_p$-stratifolds.

math.AT↗