arXiv subjects

Scotty Tilton

Publications and source records attributed to Scotty Tilton.

4 recordsLinked to original sources

arXiv2025

The Boundary Dehn Twist on a Punctured Connected Sum of Two K3 Surfaces is Nontrivial in the Smooth Mapping Class Group

We prove that the boundary Dehn twist on $K3\#K3\setminus B^4$ is nontrivial in the smooth mapping class group, providing another example of an exotic diffeomorphism on a simply-connected spin four-manifold. We do so by finding an algebraic criterion that must be satisfied if the two maps are smoothly isotopic. The main tools involved are the $\Pintwo$-equivariant families Bauer-Furuta invariant, equivariant topological $K$-theory, and the Atiyah-Hirzebruch spectral sequence to show this algebraic criterion cannot be satisfied, and this establishes the result. As a corollary, we find any smooth bundle $K3\#K3\into E\downarrow S^2$ has $w_2(T^vE)=0$, so $E$ is spin.

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arXiv2024

The $\operatorname{E}_2^{hC_6}$-homology of $\mathbb{R}P^2$ and $\mathbb{R}P^2 \wedge \mathbb{C}P^2$

Let $\operatorname{E}_2$ be the Morava E-theory of height 2 at the prime 2. In this paper, we compute the homotopy groups of $\operatorname{E}_2^{hC_6} \wedge \mathbb{R}P^2$ and $\operatorname{E}_2^{hC_6} \wedge \mathbb{R}P^2 \wedge \mathbb{C}P^2$ using the homotopy fixed point spectral sequences.

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arXiv2023

Transfer systems for rank two elementary Abelian groups: characteristic functions and matchstick games

We prove that Hill's characteristic function $\chi$ for transfer systems on a lattice $P$ surjects onto interior operators for $P$. Moreover, the fibers of $\chi$ have unique maxima which are exactly the saturated transfer systems. In order to apply this theorem in examples relevant to equivariant homotopy theory, we develop the theory of saturated transfer systems on modular lattices, ultimately producing a ``matchstick game'' that puts saturated transfer systems in bijection with certain structured subsets of covering relations. After an interlude developing a recursion for transfer systems on certain combinations of bounded posets, we apply these results to determine the full lattice of transfer systems for rank two elementary Abelian groups.

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arXiv2020

The cohomology of real Grassmannians via Schubert stratifications

In this paper, we present a closed formula for the cohomology of real Grassmannians. To achieve this, we use a theory of stratified spaces to compute the differentials in a chain complex that computes the cohomology. Specifically, we organize Schubert cells as a conically smooth stratified space in the sense of Ayala, Francis, Tanaka; the links therein yield the sought differentials, using methods in differential topology. Further, we identify the isomorphism type of this chain complex and we use this result to provide a closed formula for the additive structure of the cohomology of real Grassmannians with arbitrary coefficients.

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