# Volume 18 (2014) No. 2

## Volume 18 (2014) No. 2

### Topological complexity of 2-torsion lens spaces and $ku$-(co)homology

Donald M. Davis

**Abstract:**

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

**Abstract:**

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

**Abstract:**

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.