Volumen 18 (2014) No. 2
Volumen 18 (2014) No. 2 
Topological complexity of 2-torsion lens spaces and $ku$-(co)homology
Donald M. Davis
Resumen:
We use $ku$-cohomology to determine lower bounds for the topological complexity of mod- $2^e$ lens spaces. In the process, we give an almost complete determination of $ku_*(L^\infty (2e)) \otimes_{ku_*} ku_* (L^\infty (2e))$ proving a conjecture of González about the annihilator ideal of the bottom class. Our proof involves an elaborate row reduction of presentation matrices of arbitrary size.
Geometric dimension of stable vector bundles over spheres
Kee Yuen Lam and Duane Randall
Resumen:
We present a new method to determine the geometric dimension of stable vector bundles over spheres, using a constructive approach. The basic tools include $K$-theory and representation theory of Lie groups, and the use of spectral sequences is totally avoided.
The equivariant cohomology rings of regular nilpotent Hessenberg varieties in Lie type A: Research Announcement
Hiraku Abe, Megumi Harada, Tatsuya Horiguchi and Mikiya Masuda
Resumen:
Let $n$ be a fixed positive integer and $h: \{1,2,...,n\} \rightarrow \{1,2,...,n\}$ a Hessenberg function. The main result of this manuscript is to give a systematic method for producing an explicit presentation by generators and relations of the equivariant and ordinary cohomology rings (with $\mathbb{Q}$ coefficients) of any regular nilpotent Hessenberg variety Hess$(h)$ in type A. Specifically, we give an explicit algorithm, depending only on the Hessenberg function $h$, which produces the $n$ defining relations $\{f_{h(j),j}\}_{j=1}^n$ in the equivariant cohomology ring. Our result generalizes known results: for the case $h=(2,3,4,\ldots,n,n)$, which corresponds to the Peterson variety $Pet_n$, we recover the presentation of $H^*_S(Pet_n)$ given previously by Fukukawa, Harada, and Masuda. Moreover, in the case $h=(n,n,\ldots,n)$, for which the corresponding regular nilpotent Hessenberg variety is the full flag variety $\mathcal{Flags}(\mathbb{C}^n)$, we can explicitly relate the generators of our ideal with those in the usual Borel presentation of the cohomology ring of $\mathcal{Flags}(\mathbb{C}^n)$. The proof of our main theorem includes an argument that the restriction homomorphism $H^*_T(\mathcal{Flags}(\mathbb{C}^n)) \to H^*_S(\mbox{Hess}(h))$ is surjective. In this research announcement, we briefly recount the context and state our results; we also give a sketch of our proofs and conclude with a brief discussion of open questions. A manuscript containing more details and full proofs is forthcoming.