Mathematics

Part of the title of this article is taken from writings of Einstein, which argue that we need to exercise our ability to analyse familiar concepts, to demonstrate the conditions on which their justification and usefulness depend, and the way in which these developed, little by little … . My aim is to do this for the initial negative reactions to the seminar by E. Cech on higher homotopy groups to the ICM meeting in Z\" urich in 1932; then the subsequent work of Hurewicz, the use of groupoids and so the use of many base points, and how J.H.C. Whitehead's use of free crossed modules gave rise to a successful search for higher dimensional versions of the fundamental group and of the theorem of Van Kampen.

In [1] we introduced the notion of 'structured space', i.e. a space which locally resembles various algebraic structures. In [2] and [3] we studied some cohomology theories related to these space. In this paper we continue in this direction: while in [2] we mainly focused on cohomologies arising from f s and h , and in [3] we considered cohomologies for generalisations of objects which involved structured spaces, here we deal with (co)homologies coming from the poset associated to a structured space via an equivalence relation defined at the end of Section 4 in [1]. More precisely, we will show that various (co)homologies for posets can also be applied (under some assumptions) to structured spaces.

7 -dimensional closed and simply-connected manifolds are important objects in the theory of classical algebraic topology and differential topology. The class has been attractive since the discoveries of 7 -dimensional exotic homotopy spheres by Milnor and so on. Still recently, new understandings via algebraic topological tools such as characteristic classes and bordism relations have been discovered by Kreck, Wang, and so on. The author has been interested in understanding the class in geometric and constructive ways. Thus the author constructed explicit {\it fold} maps, which are higher dimensional versions of Morse functions, on some of the manifolds. The studies have been motivated by studies of {\it special generic} maps, higher dimensional versions of Morse functions on homotopy spheres with exactly two singular points, characterizing them topologically except 4 -dimensional cases. The class contains canonical projections of unit spheres for example. This class has been found to be interesting, restricting the topologies and the differentiable structures of the manifolds strictly: Saeki, Sakuma, Wrazidlo, and so on, found explicit phenomena. The present paper concerns fold maps on 7 -dimensional closed, simply-connected and spin manifolds whose integral cohomology rings are isomorphic to that of CP$ 2 × S 3

The mod 2 Moore spectrum C(2) is the cofiber of the self-map 2:S→S . Building on work of Burklund, Hahn, and Senger, we prove that above a line of slope 1 5 , the Adams spectral sequence for C(2) collapses at its E 5 -page and characterize the surviving classes. This completes the proof of a result of Mahowald, announced in 1970, but never proven.

We construct a model structure on the category DblCat of double categories and double functors. Unlike previous model structures for double categories, it recovers the homotopy theory of 2-categories through the horizontal embedding H:2Cat→DblCat , which is both left and right Quillen, and homotopically fully faithful. Furthermore, we show that Lack's model structure on 2Cat is both left- and right-induced along H from our model structure on DblCat . In addition, we obtain a 2Cat -enrichment of our model structure on DblCat , by using a variant of the Gray tensor product. Under certain conditions, we prove a Whitehead theorem, characterizing our weak equivalences as the double functors which admit an inverse pseudo double functor up to horizontal pseudo natural equivalence. This retrieves the Whitehead theorem for 2-categories. Analogous statements hold for the category wkDblCat s of weak double categories and strict double functors, whose homotopy theory recovers that of bicategories. Moreover, we show that the full embedding DblCat→ wkDblCat s is a Quillen equivalence.

We give a geometric method for determining the cohomology groups and the product structure of a polyhedral product, under suitable freeness conditions or with coefficients taken in a field. This is done by considering first a special class of CW pairs for which we derive a decomposition of the polyhedral product resembling a Cartan formula. The result is then generalized to arbitrary CW pairs of finite type. This leads to a direct computation of the Hilbert-Poincaré series and to other applications. The product structure on the cohomology of the polyhedral product is computed in terms of the additive generators, labelled via the Cartan decomposition. The description given suffices to enable explicit calculations.

In 1947, N.E. Steenrod defined the Steenrod Squares, which are mod 2 cohomology operations, using explicit cochain formulae for cup-i products of cocycles. He later recast the construction in more general homological terms, using group homology and acyclic model methods, rather than explicit cochain formulae, to define mod p operations for all primes p. Steenrod's student J. Adem applied the homological point of view to prove fundamental relations, known as the Adem relations, in the algebra of cohomology operations generated by the Steenrod operations. In this paper we give a proof of the mod 2 Adem relations at the cochain level. Specifically, given a mod 2 cocycle, we produce explicit cochain formulae whose coboundaries are the Adem relations among compositions of Steenrod Squares applied to the cocycle, using Steenrod's original cochain definition of the Square operations.

The construction of the Khovanov homology of links motivates an interest in decorated Boolean lattices. Placing this work in the context of a bundle theory of presheaves on small categories, we produce, for a certain set of naturally occurring cases, a Leray-Serre type spectral sequence relating the bundle to the cohomology of the total sheaf.

We construct a comparison map between the gamma homology theory of Robinson and Whitehouse and the symmetric homology theory of Fiedorowicz and Ault in the case of an augmented, commutative algebra over a unital commutative ground ring.

For G a finite group, a normalized 2-cocycle α∈ Z 2 (G, S 1 ) and X a G -space on which a normal subgroup A acts trivially, we show that the α -twisted G -equivariant K -theory of X decomposes as a direct sum of twisted equivariant K -theories of X parametrized by the orbits of an action of G on the set of irreducible α -projective representations of A . This generalizes the decomposition obtained in [Gómez J.M., Uribe B., Internat. J. Math. 28 (2017), 1750016, 23 pages, arXiv:1604.01656] for equivariant K -theory. We also explore some examples of this decomposition for the particular case of the dihedral groups D 2n with n≥2 an even integer.