arXiv subjects

Steven R. Costenoble

Publications and source records attributed to Steven R. Costenoble.

13 recordsLinked to original sources

arXiv2025

The $C_2$-equivariant ordinary cohomology of complex quadrics II: The symmetric case

In this, the second of three papers about $C_2$-equivariant complex quadrics, we calculate the equivariant ordinary cohomology of smooth symmetric quadrics graded on the representation ring of $ΠBU(1)$ and with coefficients in the Burnside Mackey functor. These calculations exhibit various interesting properties, including the first naturally occurring example we are aware of where the cohomology is not just the sum of shifted copies of the cohomology of a point, but also has summands that are shifted copies of the cohomology of the free orbit $C_2/e$.

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arXiv2025

The $C_2$-equivariant ordinary cohomology of complex quadrics III: Exceptional cases

In this, the last of three papers about $C_2$-equivariant complex quadrics, we complete the calculation of the equivariant ordinary cohomology of smooth symmetric quadrics in the cases where the fixed sets have more than two components. These calculations imply one for a $C_2$-equivariant Grassmannian, which we use to prove an equivariant refinement of the result that there are 27 lines on a cubic surface in $\mathbb{P}^3$.

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arXiv2025

The $C_2$-equivariant ordinary cohomology of complex quadrics I: The antisymmetric case

In this, the first of three papers about $C_2$-equivariant complex quadrics, we calculate the equivariant ordinary cohomology of smooth antisymmetric quadrics. One of these quadrics coincides with a $C_2$-equivariant Grassmannian, and we use this calculation to prove an equivariant refinement of the result that there are 27 lines on a cubic surface in $\mathbb{P}^3$.

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arXiv2024

The $C_2$-equivariant ordinary cohomology of $BU(2)$

We calculate the ordinary $C_2$-cohomology, with Burnside ring coefficients, of $BU(2)$, the classifying space for $C_2$-equivariant complex 2-plane bundles, using an extended grading that allows us to capture a more natural set of generators. This allows us to define characteristic classes for such bundles. Combined with earlier calculations, it also allows us to define characteristic numbers for equivariant complex lines and surfaces and we give some sample computations.

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arXiv2024

The $C_2$-equivariant ordinary cohomology of $BT^2$

We calculate the ordinary $C_2$-cohomology of $BT^2$ with Burnside ring coefficients, using an extended grading that allows us to capture a more natural set of generators. We discuss how this cohomology is related to those of $BT^1$ and $BU(2)$, calculated previously, both relationships being more complicated than in the nonequivariant case.

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arXiv2023

A geometric $C_2$-equivariant Bézout Theorem

Classically, Bézout's theorem says that an intersection of hypersurfaces in a projective space is rationally equivalent to a number of copies of a smaller projective space, the number depending on the degrees of the hypersurfaces. We give a generalization of that result to the context of $C_2$-equivariant hypersurfaces in $C_2$-equivariant linear projective space, expressing the intersection as a linear combination of equivariant Schubert varieties.

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arXiv2023

The $RO(ΠB)$-graded $C_2$-equivariant ordinary cohomology of $B_{C_2} U(1)$

We calculate the ordinary $C_2$-cohomology, with Burnside ring coefficients, of $CP_{C_2}^\infty = B_{C_2} U(1)$, the complex projective space, a model for the classifying space for $C_2$-equivariant complex line bundles. The $RO(C_2)$-graded Bredon ordinary cohomology was calculated by Gaunce Lewis, but here we extend to a larger grading in order to capture a more natural set of generators. These generators include the Euler class of the tautological bundle, which lies outside of the $RO(C_2)$-graded theory.

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arXiv2022

An algebraic $C_2$-equivariant Bézout's theorem

Bézout's theorem, nonequivariantly, can be interpreted as a calculation of the Euler class of a sum of line bundles over complex projective space, expressing it in terms of the rank of the bundle and its degree. We give here a generalization to the $C_2$-equivariant context, using the calculation of the cohomology of a $C_2$-complex projective space from an earlier paper. We use ordinary $C_2$-cohomology with Burnside ring coefficients and an extended grading necessary to define the Euler class, which we express in terms of the equivariant rank of the bundle and the degrees of the bundle and its fixed subbundles. We do similar calculations using constant $\mathbb{Z}$ coefficients and Borel cohomology and compare the results.

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arXiv2021

The $C_2$-equivariant cohomology of complex projective spaces

We compute the equivariant cohomology of complex projective spaces associated to finite-dimensional representations of $C_2$, using ordinary cohomology graded on representations of the fundamental groupoid, with coefficients in the Burnside ring Mackey functor. This extension of the $RO(C_2)$-graded theory allows for the definition of Euler classes, which are used as generators of the cohomology of the projective spaces. As an application, we give an equivariant version of Bezout's theorem.

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arXiv2017

The $\mathbb{Z}/p$ ordinary cohomology of $B_G U(1)$

With $G = \mathbb{Z}/p$, $p$ prime, we calculate the ordinary $G$-cohomology (with Burnside ring coefficients) of $\mathbb{C}P_G^\infty = B_G U(1)$, the complex projective space, a model for the classifying space for $G$-equivariant complex line bundles. The $RO(G)$-graded ordinary cohomology was calculated by Gaunce Lewis, but here we extend to a larger grading in order to capture a more natural set of generators, including the Euler class of the canonical bundle, as well as a significantly simpler set of relations.

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arXiv2017

The equivariant Spivak normal bundle and equivariant surgery for compact Lie groups

We generalize the results of a previous paper of ours to compact Lie groups. Using a recently developed ordinary equivariant homology and cohomology, we define equivariant Poincare complexes with the properties that (1) every compact G-manifold is an equivariant Poincare complex, (2) every finite equivariant Poincare complex (with some mild additional hypotheses) has an equivariant spherical Spivak normal fibration, and (3) the Pi-Pi Theorem holds for equivariant Poincare pairs under suitable gap hypotheses. The nice behavior of the ordinary equivariant homology and cohomology theories allows us to follow Wall's original line of argument closely.

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arXiv2013

Equivariant ordinary homology and cohomology

Poincare duality lies at the heart of the homological theory of manifolds. In the presence of the action of a group it is well-known that Poincare duality fails in Bredon's ordinary, integer-graded equivariant homology. We give here a detailed account of one way around this problem, which is to extend equivariant ordinary homology to a theory graded on representations of fundamental groupoids. Versions of this theory have appeared previously for actions of finite groups, but this is the first account that works for all compact Lie groups. The first part of this work is a detailed discussion of RO(G)-graded ordinary homology and cohomology, collecting scattered results and filling in gaps in the literature. In particular, we give details on change of groups and products that do not seem to have appeared elsewhere. We also discuss the relationship between ordinary homology and cohomology when the group is compact Lie, in which case the two theories are not represented by the same spectrum. The remainder of the work discusses the extension to grading on representations of fundamental groupoids, concentrating on those aspects that are not simple generalizations of the RO(G)-graded case. These theories can be viewed as defined on parametrized spaces, and then the representing objects are parametrized spectra; we use heavily foundational work of May and Sigurdsson on parametrized spectra. We end with a discussion of Poincare duality for arbitrary smooth equivariant manifolds.

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arXiv2000

Simplicial and categorical comma categories

We consider four categories: the category of diagrams of small categories indexed by a given small category O, the (comma) category of small categories over O, the category of diagrams of simplicial sets indexed by O, and the category of simplicial sets over the nerve of O. Fritsch and Golasinski claimed that these four categories have equivalent homotopy categories but, in fact, their proof contains an error and the homotopy categories are not equivalent with the weak equivalences they use in the comma categories. We show here that the correct weak equivalences are the ``weak fibre homotopy equivalences'' defined by Latch. We also construct a model category structure on the category of simplicial sets over NO in which the weak equivalences are the weak fibre homotopy equivalences.

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