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12
Quaternionic Algebraic Cycles And Reality
, 2001
"... In this paper we compute the equivariant homotopy type of spaces of algebraic cycles on real BrauerSeveri varieties, under the action of the Galois group Gal(C/R). Appropriate stabilizations of these spaces yield two equivariant spectra. The first one classifies Dupont/Seymour's quaternionic Kt ..."
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Cited by 4 (2 self)
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In this paper we compute the equivariant homotopy type of spaces of algebraic cycles on real BrauerSeveri varieties, under the action of the Galois group Gal(C/R). Appropriate stabilizations of these spaces yield two equivariant spectra. The first one classifies Dupont/Seymour's quaternionic Ktheory, and the other one classifies and equivariant cohomology theory Z # () which is a natural recipient of characteristic classes KH # (X) Z # (X) for quaternionic bundles over Real spaces X.
A freeness theorem for RO(Z/2)graded cohomology. arXiv:0908.3825v1 [math.AT
, 2009
"... Abstract. In this paper it is shown that the RO(Z/2)graded cohomology of a certain class of Rep(Z/2)complexes, which includes projective spaces and Grassmann manifolds, is always free as a module over the cohomology of a point when the coefficient Mackey functor is Z/2. ..."
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Cited by 3 (1 self)
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Abstract. In this paper it is shown that the RO(Z/2)graded cohomology of a certain class of Rep(Z/2)complexes, which includes projective spaces and Grassmann manifolds, is always free as a module over the cohomology of a point when the coefficient Mackey functor is Z/2.
Contents
, 2009
"... I am currently engaged in three distinct research projects. The areas of study are diverse but are all centered around the interplay between geometric and algebraic topology. I approach knot theory through the algebraic topology of the space of knots, and geometric data is essential in my work in in ..."
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I am currently engaged in three distinct research projects. The areas of study are diverse but are all centered around the interplay between geometric and algebraic topology. I approach knot theory through the algebraic topology of the space of knots, and geometric data is essential in my work in in group and equivariant cohomology. My dissertation work extends and refines the program of Vassiliev’s which initiated the subject of finitetype knot invariants. In order to do so, I introduce and classify plumbers ’ knots [7]. These allow me, for example, to extend the notion of the derivative of knot invariant. In [10], Paolo Salvatore, Dev Sinha and I explore the Hopf ring structue on H ∗ (BΣ•; Z/2). This structure allows us to produce a componentwise additive basis with explicit cup product multiplication rule and to understand the action of the Steenrod algebra on this basis. Lastly, Bill Kronholm and I are utilizing his RO(Z/2)graded Serre spectral sequence to classify the possible geometries of Rep(Z/2)equivariant cell complexes. In [9], we compute the equivariant cohomology of Moore spaces with coefficients in the constant Mackey functor Z. In this context, we obtain a version of his Z/2freeness theorem [12].
VANISHING THEOREMS FOR REAL ALGEBRAIC CYCLES
"... Abstract. We establish the analogue of the FriedlanderMazur conjecture for Teh’s reduced Lawson homology groups of real varieties, which says that the reduced Lawson homology of a real quasiprojective variety X vanishes in homological degrees larger than the dimension of X in all weights. As an ap ..."
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Abstract. We establish the analogue of the FriedlanderMazur conjecture for Teh’s reduced Lawson homology groups of real varieties, which says that the reduced Lawson homology of a real quasiprojective variety X vanishes in homological degrees larger than the dimension of X in all weights. As an application we obtain a vanishing of homotopy groups of the mod2 topological groups of averaged cycles and a characterization in a range of indices of dos Santos ’ real Lawson homology as the homotopy groups of the topological group of averaged cycles. We also establish an equivariant Poincare duality between equivariant FriedlanderWalker real morphic cohomology and dos Santos ’ real Lawson homology. We use this together with an equivariant extension of the mod2 BeilinsonLichtenbaum conjecture to compute some real
KERVAIRE INVARIANT ONE
"... The Kervaire sphere K 4k+1 (as described by Hirzebruch [13]) is the intersection of the complex hypersurface z 3 0 + z 2 1 + · · · + z 2 2k+1 = 0 in C2k+2 with a sphere centered at the singularity at the origin. It is a smooth manifold homeomorphic to a sphere, and it is known to bound a smooth ma ..."
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The Kervaire sphere K 4k+1 (as described by Hirzebruch [13]) is the intersection of the complex hypersurface z 3 0 + z 2 1 + · · · + z 2 2k+1 = 0 in C2k+2 with a sphere centered at the singularity at the origin. It is a smooth manifold homeomorphic to a sphere, and it is known to bound a smooth manifold W 4k+2. Attach a disk to W 4k+2 by a
THE EQUIVARIANT SLICE FILTRATION: A PRIMER
, 1107
"... 1.1. Background. Essential to the solution with Hopkins and Ravenel to the Kervaire Invariant One Problem is the construction of a new natural filtration of an equivariant spectrum, the slice filtration [5]. This is a generalization of Dugger’s slice filtration for C2equivariant homotopy theory, an ..."
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1.1. Background. Essential to the solution with Hopkins and Ravenel to the Kervaire Invariant One Problem is the construction of a new natural filtration of an equivariant spectrum, the slice filtration [5]. This is a generalization of Dugger’s slice filtration for C2equivariant homotopy theory, and it is analogous to the motivic
INVARIANT CHAINS AND THE HOMOLOGY OF QUOTIENT SPACES
, 2004
"... Abstract. For a finite group G and a finite GCWcomplex X, we construct groups H•(G, X) as the homology groups of the Ginvariants of the cellular chain complex C•(X). These groups are related to the homology of the quotient space X/G via a norm map, and therefore provide a mechanism for calculatin ..."
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Abstract. For a finite group G and a finite GCWcomplex X, we construct groups H•(G, X) as the homology groups of the Ginvariants of the cellular chain complex C•(X). These groups are related to the homology of the quotient space X/G via a norm map, and therefore provide a mechanism for calculating H•(X/G). We compute several examples and provide a new proof of “Smith theory”: if G = Z/p and X is a mod p homology sphere on which G acts, then the subcomplex X G is empty or a mod p homology sphere. We also get a new proof of the Conner conjecture: If G = Z/p acts on a Zacyclic space X, then X/G is Zacyclic. 1.
NOTES ON THE MILNOR CONJECTURES
, 2004
"... 2. Proof of the conjecture on the norm residue symbol 8 3. Proof of the conjecture on quadratic forms 15 ..."
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2. Proof of the conjecture on the norm residue symbol 8 3. Proof of the conjecture on quadratic forms 15