arXiv subjects
Publications and source records attributed to Aniruddha C. Naolekar.
A finite $CW$-complex $X$ is $C$-trivial if for every complex vector bundle $ξ$ over $X$, the total Chern class $c(ξ)=1$. In this note we completely determine when each of the following spaces are $C$-trivial: suspensions of stunted real projective spaces, suspensions of stunted complex projective spaces and suspensions of stunted quaternionic projective spaces.
Let $X^k_{m,n}=Σ^k (\mathbb R\mathbb P^m/\mathbb R\mathbb P^n)$. In this note we completely determine the values of $k,m,n$ for which the total Stiefel-Whitney class $w(ξ)=1$ for any vector bundle $ξ$ over $X^k_{m,n}$.
We define the notion of characteristic rank, $\mathrm{charrank}_X(ξ)$, of a real vector bundle $ξ$ over a connected finite $CW$-complex $X$. This is a bundle-dependent version of the notion of characteristic rank introduced by Július Korbaš in 2010. We obtain bounds for the cup length of manifolds in terms of the characteristic rank of vector bundles generalizing a theorem of Korbaš and compute the characteristic rank of vector bundles over the Dold manifolds, the Moore spaces and the stunted projective spaces amongst others.
The characteristic rank of a vector bundle $ξ$ over a finite connected $CW$-complex $X$ is by definition the largest integer $k$, $0\leq k\leq \mathrm{dim}(X)$, such that every cohomology class $x\in H^j(X;\mathbb Z_2)$, $0\leq j\leq k$, is a polynomial in the Stiefel-Whitney classes $w_i(ξ)$. In this note we compute the characteristic rank of vector bundles over the Stiefel manifold $V_k(\mathbb F^n)$, $\mathbb F=\mathbb R,\mathbb C,\mathbb H$.
Let G be any finitely generated infinite group. Denote by K(G) the FC-centre of G, i.e., the subgroup of all elements of G whose centralizers are of finite index in G. Let QI(G) denote the group of quasi-isometries of G with respect to word metric. We observe that the natural homomorphism from the group of automorphisms of G to QI(G) is a monomorphism only if K(G) equals the centre Z(G) of G. The converse holds if K(G)=Z(G) is torsion free. We apply this criterion to many interesting classes of groups.