Mapping swirls and pseudo-spines of compact 4-manifolds

AbstractA compact subset X of the interior of a compact manifold M is a pseudo-spine of M if M − X is homeomorphic to (∂M) × [0, ∞). It is proved that a 4-manifold obtained by attaching k essential 2-handles to a B3 × S1 has a pseudo-spine which is obtained by attaching k B2's to an S1 by maps of the form z → zn. This result recovers the fact that the Mazur 4-manifold has a disk pseudo-spine (which may then be shrunk to an arc). To prove this result, the mapping swirl (a “swirled” mapping cylinder) of a map to a circle is introduced, and a fundamental property of mapping swirls is established: homotopic maps to a circle have homeomorphic mapping swirls.Several conjectures concerning the existence of pseudo-spines in compact 4-manifolds are stated and discussed, including the following two related conjectures: every compact contractible 4-manifold has an arc pseudo-spine, and every compact contractible 4-manifold has a handlebody decomposition with no 3- or 4-handles. It is proved that an important class of compact contractible 4-manifolds described by Poénaru satisfies the latter conjecture

Evolution of Genetic Potential

Organisms employ a multitude of strategies to cope with the dynamical environments in which they live. Homeostasis and physiological plasticity buffer changes within the lifetime of an organism, while stochastic developmental programs and hypermutability track changes on longer timescales. An alternative long-term mechanism is “genetic potential”—a heightened sensitivity to the effects of mutation that facilitates rapid evolution to novel states. Using a transparent mathematical model, we illustrate the concept of genetic potential and show that as environmental variability decreases, the evolving population reaches three distinct steady state conditions: (1) organismal flexibility, (2) genetic potential, and (3) genetic robustness. As a specific example of this concept we examine fluctuating selection for hydrophobicity in a single amino acid. We see the same three stages, suggesting that environmental fluctuations can produce allele distributions that are distinct not only from those found under constant conditions, but also from the transient allele distributions that arise under isolated selective sweeps

A Pythagorean Theorem for Volume

Lebesgue measurable subsets A and B of parallel or identical k-dimensional affine subspaces of Euclidean n-space E^n satisfy The Product Formula for Volume: Vol_k(A)Vol_k(B) = \sum_{J \in S(n,k)} Vol_k({\pi}_J(A))Vol_k({\pi}_J(B)). Here Vol_k denotes k-dimensional Lebesgue measure; S(n,k) denotes the set of all k-element subsets of {1,2,..., n}; and for J \in S(n,k), E^J = {(x_1,x_2,...,x_n) \in E^n : x_i = 0 for all i \notin J} and {\pi}_J : E^n \rightarrow E^J is the projection that sends the i^{th} coordinate of a point of E^n to 0 whenever i \notin J. Setting B = A, we obtain the corollary: The Pythagorean Theorem for Volume: Vol_k(A)^2 = \sum_{J \in S(n,k)} (Vol_k({\pi}_J(A)))2.Comment: 19 page