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Laurence R. Taylor

Publications and source records attributed to Laurence R. Taylor.

14 recordsLinked to original sources
arXiv2022

Gauss Sums in Algebra and Topology

We consider Gauss sums associated to functions $T\to \mathbb R/\mathbb Z$ which satisfy some sort of quadratic property and investigate their elementary properties. These properties and a Gauss sum formula from the nineteenth century due to Dirichlet give the Milgram Gauss sum formula computing the signature mod $8$ of a non-singular bilinear form over $\mathbb Q$. Brown derived some results on the signature mod 8 of non-singular integral forms. Kirby and Melvin gave a formula for a generalization of this invariant to possibly non-singular forms and we further generalize it here. The Milgram Gauss sum formula and these formulas allow us to reprove Brown's result without resort to Witt group calculations. Assuming a bit of algebraic topology, we reprove a theorem of Morita's computing the signature mod $8$ of an oriented Poincaré duality space from the Pontrjagin square without using Bockstein spectral sequences. Since we work with forms which may be singular, we also obtain a version of Morita's theorem for Poincaré spaces with boundary. Finally we apply our results to the bilinear form $Sq^1x\cup y$ on $H^1(M;\mathbb Z/2\mathbb Z)$ of an orientable 3-manifold.

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arXiv2016

Surgery on Paracompact Manifolds

This is a TeX'ed version of the author's 1971 thesis from the University of California at Berkeley under the supervision of J.~B.~Wagoner.

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arXiv2015

Splitting the Kunneth formula

There is a description of the torsion product of two modules in terms of generators and relations given by Eilenberg and Mac Lane. With some additional data on the chain complexes there is a splitting of the map in the Kunneth formula in terms of these generators. Different choices of this additional data determine a natural coset reminiscent of the indeterminacy in a Massey triple product. In one class of examples the coset actually is a Massey triple product. The explicit formulas for a splitting enable proofs of results on the behavior of the interchange map and the long exact sequence boundary map on all the terms in the Kunneth formula. Information on the failure of naturality of the splitting is also obtained.

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arXiv2009

The principal fibration sequence and the second cohomotopy set

Let $p:E -> B$ be a principal fibration with classifying map $w:B -> C$. It is well-known that the group $[X,ΩC]$ acts on $[X,E]$ with orbit space the image of $p_#$, where $p_#: [X,E] -> [X,B]$. The isotropy subgroup of the map of $X$ to the base point of $E$ is also well-known to be the image of $[X, ΩB]$. The isotropy subgroups for other maps $e:X -> E$ can definitely change as $e$ does. The set of homotopy classes of lifts of $f$ to the free loop space on $B$ is a group. If $f$ has a lift to $E$, the set $p_#^{-1}(f)$ is identified with the cokernel of a natural homomorphism from this group of lifts to $[X, ΩC]$. As an example, $[X,S^2]$ is enumerated for $X$ a 4-complex.

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arXiv2008

Quadratic enhancements of surfaces: two vanishing results

This note records two results which were inexplicably omitted from our paper on Pin structures on low dimensional manifolds, [KT]. Kirby chose not to be listed as a coauthor. A Pin^- structure on a surface F induces a quadratic enhancement of the mod 2 intersection form, q: H_1(F;Z/2Z) -> Z/4Z Theorem 1.1 says that q vanishes on the kernel of the map in homology to a bounding 3-manifold. This is used by Kreck and Puppe (arXiv:0707.1599 [math.AT]) who refer for a proof to an email of the author to Kreck. A more polished and public proof seems desirable. In [KT], section 6, a Pin^- structure is constructed on a surface F dual to w_2 in an oriented 4-manifold M^4. Theorem 2.1 says that q vanishes on the Poincare dual to the image of H^1(M^4;Z/2Z) in H^1(F;Z/2Z).

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arXiv2007

Birationality of étale morphisms via surgery

We use a counting argument and surgery theory to show that if $D$ is a sufficiently general algebraic hypersurface in $\Bbb C^n$, then any local diffeomorphism $F:X \to \Bbb C^n$ of simply connected manifolds which is a $d$-sheeted cover away from $D$ has degree $d=1$ or $d=\infty$ (however all degrees $d > 1$ are possible if $F$ fails to be a local diffeomorphism at even a single point). In particular, any étale morphism $F:X \to \Bbb C^n$ of algebraic varieties which covers away from such a hypersurface $D$ must be birational.

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arXiv2006

Unoriented geometric functors

Farrell and Hsiang noticed that the geometric surgery groups defined By Wall, Chapter 9, do not have the naturality Wall claims for them. They were able to fix the problem by augmenting Wall's definitions to keep track of a line bundle. The definition of geometric Wall groups involves homology with local coefficients and these also lack Wall's claimed naturality. One would hope that a geometric bordism theory involving non-orientable manifolds would enjoy the same naturality as that enjoyed by homology with local coefficients. A setting for this naturality entirely in terms of local coefficients is presented in this paper. Applying this theory to the example of non-orientable Wall groups restores much of the elegance of Wall's original approach. Furthermore, a geometric determination of the map induced by conjugation by a group element is given.

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arXiv2005

Impossible metric conditions on exotic R^4's

There are many theorems in the differential geometry literature of the following sort. Let M be a complete Riemannian manifold with some conditions on various curvatures, diameters, volumes, etc. Then M is homotopy equivalent to a finite CW complex, or M is the interior of a compact, topological manifold with boundary. At first glance it seems unlikely that such theorems have anything to say about smooth manifolds homeomorphic to R^4. However, there is a common theme to all the proofs which forbids the existence of such metrics on most (and possibly all) exotic R^4's.

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arXiv1999

Smooth Euclidean 4-spaces with few symmetries

We say that a topologically embedded 3-sphere in a smoothing of Euclidean 4-space is a barrier provided, roughly, no diffeomorphism of the 4-manifold moves the 3-sphere off itself. In this paper we construct infinitely many one parameter families of distinct smoothings of 4-space with barrier 3-spheres. \par The existence of barriers implies, amongst other things, that the isometry group of these manifolds, in any smooth metric, is finite. In particular, S^1 can not act smoothly and effectively on any smoothing of 4-space with barrier 3-spheres.

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arXiv1998

Neighborhoods in Stratified Spaces with Two Strata

We develop a theory of tubular neighborhoods for the lower strata in manifold stratified spaces with two strata. In these topologically stratified spaces, manifold approximate fibrations and teardrops play the role that fibre bundles and mapping cylinders play in smoothly stratified spaces. Applications include the classification of neighborhood germs, the construction of exotic stratifications, a multiparameter isotopy extension theorem and an h-coborsism extension theorem.

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arXiv1998

Codimension one spheres which are null homotopic

This paper classifies embedded, codimension-one spheres which are null homotopic. This information is used to show that all null homotopic, immersed codimension-one spheres which are taut in the sense of Terng and Thorbergsson are actually distance spheres.

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arXiv1997

An invariant of smooth 4-manifolds

We define a diffeomorphism invariant of smooth 4-manifolds which we can estimate for many smoothings of R^4 and other smooth 4-manifolds. Using this invariant we can show that uncountably many smoothings of R^4 support no Stein structure. (Gompf has constructed uncountably many smoothings of R^4 which do support Stein structures.) Other applications of this invariant are given.

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