We introduce Chern classes in $U(m)$-equivariant homotopical bordism that refine the Conner–Floyd–Chern classes in the $\mathbf {MU}$-cohomology of $B U(m)$. For products of unitary groups, our Chern classes form regular sequences that generate the augmentation ideal of the equivariant bordism rings. Consequently, the Greenlees–May local homology spectral sequence collapses for products of unitary groups. We use the Chern classes to reprove the $\mathbf {MU}$-completion theorem of Greenlees–May and La Vecchia.

We consider free symmetries on cobordisms between knots, which is equivalent to cobordisms between knots in lens spaces. We classify which freely periodic knots bound equivariant surfaces in the 4-ball in terms of corresponding homology classes in lens spaces. We give a numerical condition determining the free periods for which torus knots bound equivariant surfaces in the 4-ball.

Let ${{M}^{m}}$ be an $m$-dimensional, closed and smooth manifold, equipped with a smooth involution $T\,:\,{{M}^{m}}\,\to \,{{M}^{m}}$ whose fixed point set has the form ${{F}^{n}}\,\bigcup \,{{F}^{j}}$, where ${{F}^{n}}$ and ${{F}^{j}}$ are submanifolds with dimensions $n$ and $j$, ${{F}^{j}}$ is indecomposable and $n\,>\,j$. Write $n\,-\,j\,={{2}^{p}}q$, where $q\,\ge \,1$ is odd and $p\,\ge \,0$, and set $m(n\,-\,j)\,=\,2n\,+\,p\,-q\,+\,1$ if $p\,\le \,q\,+\,1$ and $m(n\,-\,j)\,=\,2n\,+\,{{2}^{p-q}}$ if $p\,\ge \,q$. In this paper we show that $m\,\le \,m(n-j)+2j+1$. Further, we show that this bound is almost best possible, by exhibiting examples $({{M}^{m(n-j)+2j}},\,T)$ where the fixed point set of $T$ has the form ${{F}^{n}}\,\bigcup \,{{F}^{j}}$ described above, for every $2\,\le \,j\,<\,n$ and $j$ not of the form ${{2}^{t}}\,-\,1$ (for $j\,=\,0$ and 2, it has been previously shown that $m(n\,-\,j)\,+\,2j$ is the best possible bound). The existence of these bounds is guaranteed by the famous 5/2-theorem of J. Boardman, which establishes that under the above hypotheses $m\,\le \,\frac{5}{2}\,n$.

In the paper we identify up to bordism all manifolds with (Z2)k-action whose fixed point set is Sn × Sm U point.

In this paper we show that any (respectively α є Ωm) can be represented by a closed smooth (respectively closed, oriented smooth) manifold Mm admitting a smooth (Z/2)m (respectively S1)-action with a finite stationary set. We also completely determine the Grassman manifolds which are oriented boundaries as well as those which represent non-torsion elements in Ω*.