Alexander Duality and Rational Associahedra

Since Thom first introduced the notion of the ‘dual’ of a Steenrod square, in (12), it has become apparent that calculation with such duals in the cohomology of, say, a simplicial complex X will often yield information about the impossibility of embedding X in Sn, for certain values of n. For example, the celebrated theorem that cannot be embedded in can easily be proved in this way. In this paper, we seek to generalize this method to any pair of extraordinary cohomology theories h* and k*, and natural stable cohomology operation θ: h* → k*. We show in section 3 that a simplicial embeddingf: X → Sn gives rise via the Alexander duality isomorphism to a homology operationwhich is independent of n, the particular embedding f, and even the particular triangulations of X and Sn. If h* and k* are multiplicative cohomology theories, there are Kronecker productsif h0(S0) = k0(S0) = G, a field, and the Kronecker products make h*, h* and k*, k* into dual vector spaces over G, then can be turned into a cohomology operation c(θ): k*(X)→h*(X), by using this duality. This is certainly true if h* = k* = H*(;Zp), p prime, and in this case we have the original situation considered by Thom, who showed, for example, that

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Alexander duality for projections of polytopes

Topology ◽

10.1016/s0040-9383(01)00037-4 ◽

2002 ◽

Vol 41 (6) ◽

pp. 1109-1121 ◽

Cited By ~ 7

Author(s):

Xun Dong

Keyword(s):

Alexander Duality

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Random simplicial complexes, duality and the critical dimension

Journal of Topology and Analysis ◽

10.1142/s1793525320500387 ◽

2019 ◽

pp. 1-31

Author(s):

Michael Farber ◽

Lewis Mead ◽

Tahl Nowik

Keyword(s):

Simplicial Complex ◽

Knot Theory ◽

Betti Numbers ◽

Critical Dimension ◽

Simplicial Complexes ◽

Alexander Duality ◽

Random Simplicial Complexes

In this paper, we discuss two general models of random simplicial complexes which we call the lower and the upper models. We show that these models are dual to each other with respect to combinatorial Alexander duality. The behavior of the Betti numbers in the lower model is characterized by the notion of critical dimension, which was introduced by Costa and Farber in [Large random simplicial complexes III: The critical dimension, J. Knot Theory Ramifications 26 (2017) 1740010]: random simplicial complexes in the lower model are homologically approximated by a wedge of spheres of dimension equal the critical dimension. In this paper, we study the Betti numbers in the upper model and introduce new notions of critical dimension and spread. We prove that (under certain conditions) an upper random simplicial complex is homologically approximated by a wedge of spheres of the critical dimension.

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