On the Hurewicz homomorphism on the extensions of ideals in π∗s\pi_*^s and spherical classes in H∗Q0S0H_*Q_0S^0

This is about Curtis conjecture on the image of the Hurewicz map $h:{_2\pi_*}Q_0S^0\to H_*(Q_0S^0;\Z/2)$. First, we show that if $f\in{_2\pi_*^s}$ is of Adams filtration at least $3$ with $h(f)\neq 0$ then $f$ is not a decomposable element in ${_2\pi_*^s}$. Moreover, it is shown if $k$ is the least positive integer that $f$ is represented by a cycle in $\mathrm{Ext}^{k,k+n}_A(\Z/2,\Z/2)$, then (i) if $e_*h(f)\neq 0$ then $n\geqslant 2^k-1$; (ii) if $e_*h(f)=0$ then $n\geqslant 2^k-2^t$ for some $t>1$. Second, for $S\subseteq{_2\pi_{*>0}^s}$ we show that: (i) if the conjecture holds on $S$, then it holds on \la S\ra; (ii) if $h(S)=0$ then $h$ acts trivially on any extension of $S$ obtained by applying homotopy operations arising from ${_2\pi_*}D_rS^n$ with $n>0$. We also provide partial results on the extensions of \la S\ra by taking (possible) Toda brackets of its elements. We also discuss how the $EHP$-sequence information maybe applied to eliminate classes from being spherical

Generalised geometric weak conjecture on spherical classes and non-factorisation of Kervaire invariant one elements

This paper is on the Curtis conjecture. We show that the image of the Hurewicz homomorhism $h:\pi_*Q_0S^0\to H_*(Q_0S^0;\mathbb{Z})$, when restricted to product of positive dimensional elements, is determined by $\mathbb{Z}\{h(\eta^2),h(\nu^2),h(\sigma^2)\}$. Localised at $p=2$, this proves a geometric version of a result of Hung and Peterson for the Lannes-Zarati homomorphism. We apply this to show that, for $p=2$ and $G=O(1)$ or any prime $p$ and $G$ any compact Lie group with Lie algebra $\mathfrak{g}$ so that $\dim\mathfrak{g}>0$, the composition $${_p\pi_*}Q\Sigma^{n\dim\mathfrak{g}}BG_+^{\wedge n}\to {_p\pi_*}Q_0S^0\stackrel{h}{\to}H_*(Q_0S^0;\mathbb{Z}/p)$$ where $\Sigma^{n\dim\mathfrak{g}}BG_+^{\wedge n}\to S^0$ is the $n$-fold transfer, is trivial if $n>2$. Moreover, we show that for $n=2$, the image of the above composition vanishes on all elements of Adams filtration at least $1$, i.e. those elements of ${_2\pi_*^s}\Sigma^{n\dim\mathfrak{g}}BG_+^{\wedge n}$ represented by a permanent cycle $\mathrm{Ext}_{A_p}^{s,t}(\widetilde{H}^*\Sigma^{n\dim\mathfrak{g}}BG_+^{\wedge n},\mathbb{Z}/p)$ with $s>0$, map trivially under the above composition. The case of $n>2$ of the above observation proves and generalises a geometric variant of the weak conjecture on spherical classes due to Hung, later on verified by Hung and Nam. We also show that, for a compact Lie group $G$, Curtis conjecture holds if we restrict to the image of the $n$-fold transfer $\Sigma^{n\dim\mathfrak{g}}BG_+^{\wedge n}\to S^0$ with $n>1$. Finally, we show that the Kervaire invariant one elements $\theta_j\in{_2\pi_{2^{j+1}-2}^s}$ with $j>3$ do not factorise through the $n$-fold transfer $\Sigma^{n\dim\mathfrak{g}}BG_+^{\wedge n}\to S^0$ with $n>1$ for $G=O(1)$ or any compact Lie group as above.Comment: Comments are welcom

Center conditions: pull back of differential equations

The space of polynomial differential equations of a fixed degree with a center singularity has many irreducible components. We prove that pull back differential equations form an irreducible component of such a space. The method used in this article is inspired by Ilyashenko and Movasati s method. The main concepts are the Picard Lefschetz theory of a polynomial in two variables with complex coefficients, the Dynkin diagram of the polynomial and the iterated integral

Tropicalisation for Topologists

This is an attempt to look at the tropical geometry from topological point of view.Comment: A set of notes on tropical sets (no claim of originality). Comments are welcom

Towards a Browder theorem for spherical classes in ΩlSn+l\Omega^lS^{n+l}

According to Browder if $4n+2\neq 2^{t+1}-2$ then the Kervaire invariant of the cobordism class of a $(4n+2)$-dimensional manifold $M^{4n+2}$ vanishes and $M^{2^{t+1}-2}$ is of Kervaire invariant one if and only if $h_t^2\in\mathrm{Ext}(\mathbb{Z}/2,\mathbb{Z}/2)$ is a permanent cycle. On the other hand, according to Madsen if $4n+2\neq 2^t-2$ then $M^{4n+2}$ is cobordant to a sphere (hence of Kervaire invariant zero) and $M^{2^{t+1}-2}$ is not cobordant to a sphere (hence of Kervaire invariant one) if and only if certain element $p_{2^{t}-1}^2\in H_*QS^0$ is spherical. Moreover, it is known that $p_{2^t-1}^2$ is spherical if and only if $h_t^2$ is a permanent cycle in the Adams spectral sequence. Moreover, classes $p_{2n+1}^2\in H_*QS^0$ with $2n+1\neq 2^t-1$ are easily eliminated from being spherical. Hence, Browder's theorem admits a presentation and proof in terms of certain square classes being spherical in $H_*QS^0$ (see also work of Akhmetev and Eccles). In this note, we consider the problem of determining spherical classes $H_*\Omega^lS^{n+l}$ with $n>0$ and $4\leqslant l\leqslant +\infty$. We show (1) if $\xi^2\in H_*\Omega^lS^{n+l}$ is given with $\dim\xi+1\neq 2^t$ and $\dim \xi+1\equiv 2\textrm{ mod }4$ and $n>l$, then $\xi^2$ is not spherical. We refer to this as a generalised Browder theorem. We also present some partial results on the degenerate cases, corresponding to $\dim\xi\neq 2^t-1$, when $l>n$. (2) For $l\in\{4,5,6,7,8\}$ the only spherical classes in $H_*\Omega^lS^{n+l}$ arise from the inclusion of the bottom cell, or the Hopf invariant one elements. This verifies Eccles conjecture when restricted to finite loop spaces with $l<9$.Comment: Comments are welcom

Filtered finiteness of the image of the unstable Hurewicz homomorphism with applications to bordism of immersions

After recent work of Hill, Hopkins, and Ravenel on the Kervaire invariant one problem, as well as Adams' solution of the Hopf invariant one problem, an immediate consequence of Curtis conjecture is that the set of spherical classes in $H_*Q_0S^0$ is finite. Similarly, Eccles conjecture, when specialised to $X=S^n$ with $n>0$, together with Adams' Hopf invariant one theorem, implies that the set of spherical classes in $H_*QS^n$ is finite. We prove a filtered version of the above the finiteness properties. We show that if $X$ is an arbitrary $CW$-complex such that $H_*X$ is finite dimensional then the image of the composition ${_2\pi_*}\Omega^l\Sigma^{l+2}X\to{_2\pi_*}Q\Sigma^2X\to H_*Q\Sigma^2X$ is finite; the finiteness remains valid if we formally replace $X$ with $S^{-1}$. As an immediate and interesting application, we observe that for any compact Lie group $G$ with $\dim\mathfrak{g}>1$ and for any $n>0$ the image of the composition ${_2\pi_*}Q\Sigma^{\dim\mathfrak{g}}BG_+^{[n]}\to{_2\pi_*}Q\Sigma^{\dim\mathfrak{g}}BG_+\to {_2\pi_*}Q_0S^0\to H_*Q_0S^0$ is finite where $\Sigma^{\dim\mathfrak{g}}BG_+\to S^0$ is a suitably twisted transfer map. Next, we consider work of Koschorke and Sanderson which using Thom-Pontrjagin construction provides a $1$-$1$ correspondence (a group isomorphism if $m+d>0$) $\Phi^{N,\xi}_{m,d}: \mathrm{Imm}_\xi^d(\mathbb{R}^m\times N) \longrightarrow [N_+,\Omega^{m+d}\Sigma^dT(\xi)]$. We apply work of Asadi and Eccles on computing Stiefel-Whitney numbers of immersions to show that given a framed immersion $M\to\mathbb{R}^{n+k}$ and choosing $n$ very large with respect to $d$ and $k$, all self-intersection manifolds of an arbitrary element of $\mathrm{Imm}_\xi^d(\mathbb{R}^m\times N)$ are boundary.Comment: This is the submitted version of this work. Comments are welcome! A mis-statement in Corollary 1.7 is corrected, and the abstract is adopted accordingl

Estimating Target Signatures with Diverse Density

Hyperspectral target detection algorithms rely on knowing the desired target signature in advance. However, obtaining an effective target signature can be difficult; signatures obtained from laboratory measurements or hand-spectrometers in the field may not transfer to airborne imagery effectively. One approach to dealing with this difficulty is to learn an effective target signature from training data. An approach for learning target signatures from training data is presented. The proposed approach addresses uncertainty and imprecision in groundtruth in the training data using a multiple instance learning, diverse density (DD) based objective function. After learning the target signature given data with uncertain and imprecise groundtruth, target detection can be applied on test data. Results are shown on simulated and real data.Comment: Appeared in Proceedings of the 7th IEEE Workshop on Hyperspectral Image and Signal Processing: Evolution in Remote Sensing, Tokyo, Japan, 201

Splitting Madsen-Tillmann spectra II. The Steinberg idempotents and Whitehead conjecture

We show that, at the prime $p=2$, the spectrum $\Sigma^{-n}D(n)$ splits off the Madsen-Tillmann spectrum $MTO(n)=BO(n)^{-\gamma_n}$ which is compatible with the classic splitting of $M(n)$ off $BO(n)_+$. For $n=2$, together with our previous splitting result on Madsen-Tillmann spectra, this shows that $MTO(2)$ is homotopy equivalent to $BSO(3)_+\vee\Sigma^{-2}D(2)$.Comment: Comments are welcome

Multiple Instance Hybrid Estimator for Learning Target Signatures

Signature-based detectors for hyperspectral target detection rely on knowing the specific target signature in advance. However, target signature are often difficult or impossible to obtain. Furthermore, common methods for obtaining target signatures, such as from laboratory measurements or manual selection from an image scene, usually do not capture the discriminative features of target class. In this paper, an approach for estimating a discriminative target signature from imprecise labels is presented. The proposed approach maximizes the response of the hybrid sub-pixel detector within a multiple instance learning framework and estimates a set of discriminative target signatures. After learning target signatures, any signature based detector can then be applied on test data. Both simulated and real hyperspectral target detection experiments are shown to illustrate the effectiveness of the method

Hyperspectral Unmixing with Endmember Variability using Partial Membership Latent Dirichlet Allocation

The application of Partial Membership Latent Dirichlet Allocation(PM-LDA) for hyperspectral endmember estimation and spectral unmixing is presented. PM-LDA provides a model for a hyperspectral image analysis that accounts for spectral variability and incorporates spatial information through the use of superpixel-based 'documents.' In our application of PM-LDA, we employ the Normal Compositional Model in which endmembers are represented as Normal distributions to account for spectral variability and proportion vectors are modeled as random variables governed by a Dirichlet distribution. The use of the Dirichlet distribution enforces positivity and sum-to-one constraints on the proportion values. Algorithm results on real hyperspectral data indicate that PM-LDA produces endmember distributions that represent the ground truth classes and their associated variability