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We establish some structural results for the Witt and Grothendieck--Witt groups of schemes over \Z[1/2], including homotopy invariance for Witt groups and a formula for the Witt and Grothendieck--Witt groups of punctured affine spaces over a scheme. All these results hold for singular schemes and at the level of spectra. 

Abstract This work is motivated by the following question in data-driven study of dynamical systems: given a dynamical system that is observed via time series of persistence diagrams that encode topological features of snapshots of solutions, what conclusions can be drawn about solutions of the original dynamical system? We address this challenge in the context of an N dimensional system of ordinary differential equation defined in $${\mathbb {R}}^N$$ R N . To each point in $${\mathbb {R}}^N$$ R N (e.g. an initial condition) we associate a persistence diagram. The main result of this paper is that under this association the preimage of every persistence diagram is contractible. As an application we provide conditions under which multiple time series of persistence diagrams can be used to conclude the existence of a fixed point of the differential equation that generates the time series.