The algebra of higher homotopy operations
We explain how the simplicial higher-order unstable homotopy operations defined in [BBS2] may be composed and inserted one in another, thus forming a coherent if complicated algebraic structure.
math.AT↗
arXiv subjects
Debasis Sen
Publications and source records attributed to Debasis Sen.
Monoids related to self homotopy equivalences of fibred product
We introduce a nested sequence of monoids related to self-homotopy equivalences of fibrewise pointed spaces, such that the limit is the group of homotopy classes of fibrewise pointed self-equivalences. We explore this monoid for the fibred product in terms of individual spaces. Further we study two related invariants associated to these monoids: self closeness number and self length.
math.AT↗
Equivariant self-homotopy equivalences of product spaces
Let G be a finite group. We study the group of G-equivariant self-homotopy equivalences of product of G-spaces. For a product of n-spaces, we represent it as product of n-subgroups under the assumption of equivariant reducibility. Further we describe each factor as a split short exact sequence. Also, we obtain an another kind of factorisation, called $LU$ type decomposition, as product of two subgroups.
math.AT↗
Nucleation-growth versus spinodal decomposition in Fe-Cr alloys: an experimental verification by atom probe tomography and small angle neutron scattering
Identifying the operative mode of phase separation (spinodal decomposition (SD) or nucleation-growth (NG)) remains a largely unexplored area of research in spite of its importance. The present work examines this critically in Fe-Cr system using atom probe tomography (APT) and small angle neutron scattering (SANS), and establishes the framework to distinguish the two different modes of alpha-prime phase separation in thermally aged Fe-35 at.% Cr and Fe-20 at.% Cr alloys. Independent APT analysis determines the mode of phase separation on the basis of: (i) presence / absence of periodic chemical fluctuation through radial distribution function analysis; and (ii) inter-phase interface characteristics (diffuse / sharp). SANS analysis, in contrast, yields virtually indistinguishable correlation peaks for both the modes, which necessitates further investigation of the several different aspects of SANS profiles in the light of APT results. For the first time, key features of SANS profiles have been identified that can unambiguously distinguish SD from NG in Fe-Cr system: (i) nature of temporal evolution of FWHM of the correlation peak; and (ii) appropriate value of 'gamma' for fitting with the dynamic scaling model ('gamma'= 6 for SD, Fe-35 at.% Cr alloy; 'gamma'= 4 for NG, Fe-20 at.% Cr alloy).
cond-mat.mtrl-sci↗
Simplicial and combinatorial versions of higher symmetric topological complexity
In this paper, we introduce higher symmetric simplicial complexity $SC_n^{\Sigma}(K)$ of a simplicial complex $K$ and higher symmetric combinatorial complexity $CC_n^{\Sigma}(P)$ of a finite poset $P$. These are simplicial and combinatorial approaches to symmetric motion planning of Basabe - Gonz\'{a}lez - Rudyak - Tamaki. We prove that the symmetric simplicial complexity $SC_n^{\Sigma}(K)$ is equal to symmetric topological complexity $TC_n^{\Sigma}(|K|)$ of the geometric realization of $K$ and the symmetric combinatorial complexity $CC_n^{\Sigma}(P)$ is equal to symmetric topological complexity $TC_n^{\Sigma}(|\mathcal{K}(P)|)$ of the geometric realization of the order complex of $P$.
math.AT↗
The higher structure of unstable homotopy groups
We construct certain unstable higher-order homotopy operations indexed by the simplex categories of $\Delta^{n}$ for ${n\geq 2}$ and prove that all elements in the homotopy groups of a wedge of spheres are generated under such operations by Whitehead products and the group structure. This provides a stronger unstable analogue of Cohen's theorem on the decomposition of stable homotopy.
math.AT↗
An upper bound for higher topological complexity and higher strongly equivariant complexity
We prove an upper bound of higher topological complexity $TC_n(X)$ using higher $\mathcal{D}$-topological complexity $TC_n^{\mathcal{D}}(X)$ of a space $X$. An intermediate invariant $\widetilde{TC}_n(X)$ is used in the proof. We interpret this invariant $\widetilde{TC}_n(X)$ as higher analogue of strongly equivariant topological complexity of the universal cover of $\widetilde{X}$ with the action of the fundamental group of $X$.
math.AT↗
Magnetic ordering of the martensite phase in Ni-Co-Mn-Sn-based ferromagnetic shape memory alloys
The magnetic state of low temperature martensite phase in Co-substituted Ni-Mn-Sn-based ferromagnetic shape memory alloys (FSMAs) has been investigated, in view of numerous conflicting reports of occurrences of spin glass (SG), superparamagnetism (SPM) or long range anti-ferromagnetic (AF) ordering. Combination of dc magnetization, ac susceptibility and small angle neutron scattering (SANS) studies provide a clear evidence for AF order in martensitic phase of Ni45Co5Mn38Sn12 alloy and rule out SPM and SG orders. Identical studies on another alloy of close composition of Ni44Co6Mn40Sn10 point to presence of SG order in martensitic phase and absence of SPM behavior, contrary to earlier report. SANS results do show presence of nanometre-sized clusters but they are found to grow in size from 3 nm at 30 K to 11 nm at 300 K, and do not correlate with magnetism in these alloys.
cond-mat.mtrl-sci↗
Magnetic memory effect in ensembles of interacting anisotropic magnetic nanoparticles
We explore the influence of demagnetization interaction on magnetic memory effect by varying organization geometry of anisotropic ZnFe$_2$O$_4$ nanoparticles in an ensemble. The static and dynamic behaviour of two differently organized ensembles, compact ensemble (CE) and hollow core ensemble (HCE), are extensively studied by both dc and ac susceptibility, magnetic memory effect and spin relaxation. The frequency-dependence peak shifting of freezing temperature in both the systems is analyzed properly with the help of two dynamic scaling models: Vogel-Fulcher law and power law. Presence of cluster spin-glass phase is reflected from Vogel-Fulcher temperature $T_0$ $\simeq$ 142.58 K for CE, $\simeq$ 97 K for HCE and characteristic time constant $\tau_0$ $\simeq$ $8.85\times10^{-9}$ s for CE, $\simeq$ $3.8\times10^{-10}$ s for HCE; along with $\delta$T$_{Th}$ $\sim$ 0.1 for CE and 0.2 for HCE. The power law fitting with dynamic exponent $zv'$ = 6.2 $\pm$ 1.1 for CE, 6.3 $\pm$ 0.5 for HCE and single spin flip $\tau^*$ $\simeq$ $7.7\times10^{-11}$ s for CE, $\simeq$ $1.3\times10^{-10}$ s for HCE provide firm confirmation of cluster spin-glass phase. The progressive spin freezing across multiple metastable states with prominent memory effects is reflected in both the systems via nonequilibrium dynamics study. The hollow core geometry with anisotropic nanoparticles on surface with closer proximity leads to complex anisotropy energy landscape with enhanced demagnetizing field resulting highly frustrated surface spins. As a consequence, more prominent magnetic memory effect is observed in HCE with higher activation energy, reduced blocking temperature and enhanced coercivity than that of CE.
cond-mat.mtrl-sci↗
Note on Toda brackets
We provide a general definition of Toda brackets in a pointed model categories, show how they serve as obstructions to rectification, and explain their relation to the classical stable operations.
math.AT↗
Higher structure in the unstable Adams spectral sequence
We describe a variant construction of the unstable Adams spectral the sequence for a space $Y$, associated to any free simplicial resolution of $H^*(Y;R)$ for $R=\mathbb{F}_p$ or $\mathbb{Q}$. We use this construction to describe the differentials and filtration in the spectral sequence in terms of appropriate systems of higher cohomology operations.
math.AT↗
Higher cohomology operations and R-completion
Let $R=\mathbb{F}_p$ or a field of characteristic $0$. For each $R$-good topological space $Y$, we define a collection of higher cohomology operations which, together with the cohomology algebra $H^*(Y;R)$ suffice to determine $Y$ up to $R$-completion. We also provide a similar collection of higher cohomology operations which determine when two maps $f_0,f_1: Z\to Y$ between $R$-good spaces(inducing the same algebraic homomorphism $H^*(Y;R)\to H^*(Z;R)$) are $R$-equivalent.
math.AT↗
Finite group actions on Kan complexes
We study simplicial action of groups on one vertex Kan complexes. We show that every semi-direct product of the fundamental group of an one vertex Kan complex with a finite group can be simplicially realized. We also calculate the cohomology of the fixed point set of a finite $p-$group action on an one vertex aspherical Kan complex.
math.AT↗
Mapping spaces and R-completion
We study the questions of how to recognize when a simplicial set X is of the form X=map(Y,A) for a given space A, and how to recover Y from X, if so. A full answer is provided when A=K(R,n), for $R=\mathbb{F}_p$ or $\mathbb{Q}$, in terms of a mapping algebra structure on X (defined in terms of product-preserving simplicial functors out of a certain simplicially-enriched sketch). In addition, when A is a suitable infinite loop space for a suitable connective ring spectrum, we can recover Y from map(Y,A) given such a mapping algebra structure. Most importantly, our methods provide a new way of looking at the classical Bousfield-Kan R-completion.
math.AT↗
Realizing homotopy group actions
For any finite group $G$, we define the notion of a Bredon homotopy action of $G$, modelled on the diagram of fixed point sets $(X_H)_{H\leq G}$ for a $G$-space $X$, together with a pointed homotopy action of the group $N_{G}H/H$ on $X^{H}/(\bigcup_{H<K} X^{K})$. We then describe a procedure for constructing a suitable diagram $\underline{X}:O_G^{op}\to Top$ from this data, by solving a sequence of elementary lifting problems. If successful, we obtain a $G$-space $X'$ realizing the given homotopy information, determined up to Bredon $G$-homotopy type. Such lifting methods may also be used to understand other homotopy questions about group actions, such as transferring a $G$-action along a map $f:X\to Y$.
math.AT↗
Representing Bredon cohomology with local coefficients
For a discrete group G, we represent the Bredon cohomology with local coefficients as the homotopy classes of maps in the category of equivaraint crossed complexes. Subsequently, we construct a naive parametrized G-spectrum, such that the cohomology theory defined by it reduces to the Bredon cohomology with local coefficients when restricted to suspension spectra.
math.AT↗
A spectral sequence in Bredon cohomology
For any finite group G, we construct a spectral sequence for computing the Bredon cohomology of a G-CW complex X, starting with the cohomology of X^H/\cup_{K>H}X^K with suitable local coefficients, for various H \leq G.
math.AT↗
A Generalized Goursat Lemma
In this note the usual Goursat lemma, which describes subgroups of the direct product of two groups, is generalized to describing subgroups of a direct product $A_1\times A_2 \times...\times A_n$ of a finite number of groups. Other possible generalizations are discussed and applications characterizing several types of subgroups are given. Most of these applications are straightforward, while somewhat deeper applications occur in the case of profinite groups, cyclic groups, and the Sylow $p$-subgroups (including infinite groups that are virtual $p$-groups).
math.GR↗