Mathematics

7-dimensional closed and simply-connected manifolds have been attractive as central and explicit objects in algebraic topology and differential topology of higher dimensional closed and simply-connected manifolds, which were studied actively especially in the 1950s--60s. Attractive studies of the class of these 7 -dimensional manifolds were started by the discovery of so-called exotic spheres by Milnor. It has influenced on the understanding of higher dimensional closed and simply-connected manifolds via algebraic and abstract objects. Recently this class is studied via more concrete notions from algebraic topology such as concrete bordism theory by Crowley, Kreck, and so on. As a new kind of fundamental and important studies, the author has been challenging understanding the class in constructive ways via construction of fold maps, which are higher dimensional versions of Morse functions. The present paper presents a new general method to construct ones on spin manifolds of the class.

Motivated by the philosophy that C ??-algebras reflect noncommutative topology, we investigate the stable homotopy theory of the (opposite) category of C ??-algebras. We focus on C ??-algebras which are non-commutative CW-complexes in the sense of [ELP]. We construct the stable ??-category of noncommutative CW-spectra, which we denote by NSp . Let M be the full spectral subcategory of NSp spanned by "noncommutative suspension spectra" of matrix algebras. Our main result is that NSp is equivalent to the ??-category of spectral presheaves on M . To prove this we first prove a general result which states that any compactly generated stable ??-category is naturally equivalent to the ??-category of spectral presheaves on a full spectral subcategory spanned by a set of compact generators. This is an ??-categorical version of a result by Schwede and Shipley [ScSh1]. In proving this we use the language of enriched ??-categories as developed by Hinich [Hin2,Hin3]. We end by presenting a "strict" model for M . That is, we define a category M s strictly enriched in a certain monoidal model category of spectra S p M . We give a direct proof that the category of S p M -enriched presheaves M op s ?�S p M with the projective model structure models NSp and conclude that M s is a strict model for M .

We provide a new method to compute the (homotopy) fixed-points of the permutation action on H F 2 ∧H F 2 by relating it to Real bordism. More precisely, we identify the C 4 -pullback of the C 2 -spectrum N C 2 e H F 2 with a localization of N C 4 C 2 M U R . This allows us to use the localized slice spectral sequence for the computation of π C 2 ★ N C 2 e H F 2 . From this, we compute the first eight homotopy groups and deduce an infinite family of differentials in the homotopy fixed point spectral sequence.

We determine (non-)triviality of Samelson products of inclusions of factors of the mod p decomposition of G (p) for (G,p)=( E 7 ,5),( E 7 ,7),( E 8 ,7) . This completes the determination of (non-)triviality of those Samelson products in p -localized exceptional Lie groups when G has p -torsion free homology.

We compare two notions of G -fiber bundles and G -principal bundles in the literature, with an aim to clarify early results in equivariant bundle theory that are needed in current work of equivariant algebraic topology. We also give proofs of some equivariant generalizations of well-known non-equivariant results involving the classifying space.

We give an obstruction for lifts and extensions in a model category inspired by Klein and Williams' work on intersection theory. In contrast to the familiar obstructions from algebraic topology, this approach produces a single invariant that is complete in the presences of the appropriate generalizations of dimension and connectivity assumptions.

We define, in C p -equivariant homotopy theory for p>2 , a notion of μ p -orientation analogous to a C 2 -equivariant Real orientation. The definition hinges on a C p -space CP ∞ μ p , which we prove to be homologically even in a sense generalizing recent C 2 -equivariant work on conjugation spaces. We prove that the height p−1 Morava E -theory is μ p -oriented and that tmf(2) is μ 3 -oriented. We explain how a single equivariant map v μ p 1 : S 2ρ → Σ ∞ CP ∞ μ p completely generates the homotopy of E p−1 and tmf(2) , expressing a height-shifting phenomenon pervasive in equivariant chromatic homotopy theory.

The study of the topology of the superlevel sets of stochastic processes on [0,1] in probability led to the introduction of $\R$-trees which characterize the connected components of the superlevel-sets. We provide a generalization of this construction to more general deterministic continuous functions on some path-connected, compact topological set X and reconcile the probabilistic approach with the traditional methods of persistent homology. We provide an algorithm which functorially links the tree T f associated to a function f and study some invariants of these trees, which in 1D turn out to be linked to the regularity of f .

The Alexandroff-?ech normal cohomology theory [Mor 1 ], [Bar], [Ba 1 ],[Ba 2 ] is the unique continuous extension \cite{Wat} of the additive cohomology theory [Mil], [Ber-Mdz 1 ] from the category of polyhedral pairs K 2 Pol to the category of closed normally embedded, the so called, P -pairs of general topological spaces K 2 Top . In this paper we define the Alexander-Spanier normal cohomology theory based on all normal coverings and show that it is isomorphic to the Alexandroff-?ech normal cohomology. Using this fact and methods developed in [Ber-Mdz 3 ] we construct an exact, the so called, Alexander-Spanier normal homology theory on the category K 2 Top , which is isomorphic to the Steenrod homology theory on the subcategory of compact pairs K 2 C . Moreover, we give an axiomatic characterization of the constructed homology theory.

The loop product is an operation in string topology. Cohen and Jones gave a homotopy theoretic realization of the loop product as a classical ring spectrum L M −TM for a manifold M . Using this, they presented a proof of the statement that the loop product is isomorphic to the Gerstenhaber cup product on the Hochschild cohomology H H ∗ ( C ∗ (M); C ∗ (M)) for simply connected M . However, some parts of their proof is technically difficult to justify. The main aim of the present paper is to give detailed modification to a geometric part of their proof. To do so, we set up an "up to higher homotopy" version of McClure-Smith's cosimplicial product. We prove a structured version of Cohen-Jones isomorphism in the category of symmetric spectra.