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Questions tagged [spectra]

Use this tag for question concerning the notion of spectra in algebraic topology. Spectra are natural objects which arises when dealing with stable properties. For spectra of linear operators, please use the tag (spectral-theory) instead.

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Image of a spectra map to a module spectrum

Let $X$ be a symmetric spectrum and $M$ be an $E$-module for some ring spectrum $E$. Consider a spectra map $f: X \to M.$ Then the image $Im(f)$ may not be an $E$-submodule of $M$ ( even may not be ...
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Cohomology of spectra as free modules over Steenrod Algebra

I'm studying the construction of Adams spectral sequence in Hatcher's "Algebraic Topology" chapter five; I'm in trouble with an "algebraic statement". Let $X$ be a connective CW-spectrum of finite ...
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30 views

Tate construction of the Anderson dual of the sphere spectrum

I am trying to understand example I.2.3 of the article "On Topological Cyclic Homology", i.e. im trying to see the necesity of the bounded below assumption in the Tate Orbit Lemma (Lemma I.2.1) by ...
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1answer
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if $\lambda \in sp(u)$ why $\lambda \in sp(\phi)$ too?

$E$ est un $K-$espace vectoriel de dimension finie $n$ , et u $ \in L(E) $ et $\phi$ une application de $L(E)$ vers $L(E)$ et soit $\phi(v)=u \circ v$ Montrons que si $\lambda \in sp(u)$ alors $\...
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67 views

Does trivial cohomology of spectra imply trivial homology?

It is known that for any spectrum $X$, $H\mathbb{Z}^*(X)=0$ implies that $H\mathbb{Z} \wedge X =0$. Also, for the case $HF_p,$ if we consider $ HF_p^*(X) =0.$ This gives $[HF _ p \wedge X , \sum^i ...
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38 views

Spectral decomposition of an operator

Given the operator: $$\begin{pmatrix} i & 0 & - 4\\ 0 & - 3i & 0\\ 2 & 0 & - i \end{pmatrix}$$ Now $det(\lambda1-A)=(z+3i)(z^2+9)=(z+3i)^2(z-3i)$, I know a theorem that says ...
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How is a spectra function related to the mean value and standard deviation?

everyone. I'm starting to research stochastic optimization algorithms, and my simulation model is an aircraft which endures turbulence. I've read a document on the atmosphere environment provided by ...
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14 views

hermitian operators with the same spectra

I want to prove that if two hermitian matrices say $H_1$ and $H_2$ have the same spectra, then one of them is conjugate of the other by some unitary matrix. $$H_1 = U H_2 U^{\dagger}$$ Any idea?
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74 views

Structure of module over Eilenberg MacLane spectrum

Let $HR$ be the Eilenberg-Maclane spectrum for a commutative ring $R$ and $M$ be a module over $HR.$ Then I want to prove that $M$ is a product of Eilenberg-Mac Lane spectra. Construction: Let $\...
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64 views

properties of the spectrum in predicate logic

I want to proof that the set of spectra is closed under union and intersection. I know that it works somehow by assuming that the languages of the two formulas $\phi_1$ and $\phi_2$ are disjoint and ...
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26 views

The essential spectram of the compact manifold

I heard that the essential spectrum of the compact Riemannian manifold is empty. Why does this hold? I would appreciate if you could teach me about this.
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39 views

Loops Infinity of a spectrum

Let $\mathbf{X}$ be an (orthogonal) spectrum (can assume that it's an $\Omega$-spectrum if this helps give a positive answer) and give the category of orthogonal spectra the stable model structure. ...
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1answer
128 views

Convenient categories in algebraic topology: their importance, and the role topology plays in their construction

Disclaimer. I have stated three questions but I felt that they are so related that they fit within a single post. Context. After reading Hatcher's Algebraic Topology I wanted to learn more about ...
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70 views

Construction of $(p)$-local spectrum.

I have a question regarding the construction of the $p$-local spectrum of a given spectrum $X$. I have seen in many papers people defining the dual notion of it, namely the $p$-completion of $X$, $X_p^...
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1answer
79 views

Definition of the Berkovich spectrum

I am trying to read these notes: http://www-personal.umich.edu/~takumim/Berkovich.pdf Regarding the Berkovich spectrum. In definition [2.24] it says that the spectrum is the set of bounded (non-...
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1answer
72 views

Realizing the Berkovich affine line as a union of Berkovich spectrums

I am trying to understand what is the relation of the affine Berkovich space to the Berkovich space on an appropriate polynomial ring. A more exact version of the question is as follows: Let $(K,\...
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35 views

Relation of $\mathbb{Z}_2$-cohomology and interger cohomology

In the last chapter of May's Concise course, we have an isomorphism of $\mathbb{Z}_2$-cohomology of spectra, induced by a map of spectra. And then May writes, it can be deduced that the induced map on ...
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Homotopy of wedge sum

In the last chapter of his Concise course in algebraic topology, May states (without proof or reference) that for an arbitrary collection $(X_i)_{i\in I}$ of spectra the following hold: $\pi_n(\prod_{...
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19 views

Meaning of cocycle on a spectrum?

What is the meaning of the notation $Z^{p}(F,\mathbb{R})$ where $F$ is a spectrum ? I encountered this in a paper (see the line after remark 4.49) that I am reading, but can not understand the meaning....
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176 views

A question regarding generalized cohomology and spectra : proof of $E^{\ast}(S)\otimes\mathbb{R} = H^{\ast}(S;\pi_{\ast}E\otimes \mathbb{R})$

I am reading the paper Quadratic functions in geometry, topology and M theory by M.J.Hopkins and I.M.Singer, and in section 4.8 they say : 'Recall that for any compact $S$, and for any cohomology ...
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41 views

spectrum theory in generalized cohomology

why is generalized cohomology theory equivalent to spectrum theory ? specifically, why is that any generalized cohomology theory is represented by a spectrum and a spectrum gives a ...
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56 views

Reference for spectra theory (in topology)

Is there any good reference for the theory of spectra (in topology)? Thanks a lot for your help!
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38 views

Isomorphism in the case of real symmetric matrices

I'm looking for clarification of the following problem: it's known that determining which graphs are uniquely determined by their spectra is in general a very hard problem. But what about more ...
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Reference request: Representability of multiplicative equivariant cohomology theories

Let $G$ be a topological group, say a compact Lie group, and $e^*_G$ a multiplicative $\mathbb Z$-graded $G$-equivariant cohomology theory defined on $G$–CW complexes. Is there some analogue result to ...
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1answer
186 views

Is there a stable Hurewicz Theorem?

Given a $(n-1)$-connected spectrum $E$ , is the natural morphism ${\pi _k}\left( E \right) \to {\pi _k}\left( {H\mathbb{Z} \wedge E} \right)$ an isomorphism for $k \leq n$? I think yes, but I can't ...
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1answer
140 views

Finite extension of Noetherian rings [duplicate]

I've got two noetherian rings $A\subset B$ such that $B$ is a finite $A$-module. Now, if I consider the associated map between spectra that given $q \in \operatorname{Spec} B$ consider $q \cap A \in \...
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34 views

Unexplained Mathematical Passage in Fourier Series

just as an FYI, this is an exercise from Digital Signal Processing by Proakis & Manolakis 4th Edition. I need to calculate the spectrum X(F) of a signal given by a graph. That part was easy and I ...
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1answer
87 views

Spectral realization of the natural transformation from bordism to singular homology

There is a geometric version of the definition of singular homology $H_n$ in terms of continuous maps of "pseudomanifolds" ($n$-dimensional simplicial complexes such that every $(n-1)$-simplex is ...
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1answer
130 views

Reference request: Adjunction of $\Sigma^\infty$ and $\Omega^\infty$ is monoidal

I would like to see that the map of spectra $\Sigma^\infty \mathbb{C}P^\infty_+ \to KU$ is actually a map of ring spectra, where $KU$ denotes the complex K-theory spectrum and the map is given as the ...
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1answer
74 views

Splitting of a spectrum as a wedge

Suppose that $i:E\to F$ and $r:F\to E$ are maps of spectra ($S^1$-spectra of topological spaces) such that $r\circ i$ is a homotopy equivalence. Can we always show that the spectrum $F$ splits as a ...
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Intuition behind spectra of products of reflections

I was reading this article by Szegedy Quantum Speed-Up of Markov Chain based Algorithms, and understand the calculations. Yet I am missing an intuition about why the reflection about the column-space ...
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43 views

Prove that the spectrum of an operator on L^p (\mathbb{R}) is the boundary of unit cỉrcle

I am working on finding the spectrum of some bounded linear operator but I have no idea. Could you please help me to prove the following problem: Let $X=L^p (\mathbb{R})$ and for any $t \in \mathbb{R}...
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1answer
581 views

The spectrum of a bounded linear operator on a $X=C[0;1]$

I am working with the book "Introductory Functional Analysis With Applications", written by Kreyzig and I got the trouble with problem 1 and problem 2 in section 7.3, which are: Problem 1: "Let $X=C[...
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62 views

Defining coefficient ring of a cohomology theory

Let $E$ be a ring spectrum with multiplication given by $\mu$. Then we make $E^*(X)$ a module over $E^*(pt)$ as follows. We give a map $$\phi: [S^{-n},E] \times [\Sigma^{-m}X,E] \to [\Sigma^{-m-n}X,E ]...
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2answers
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Calculating the extraordinary cohomology of $\mathbb{C}P^n$

Let $E$ be a ring spectrum with an orientation. Now I want to calculate $E^*(\mathbb{C}P^n)$. The definition of orientation I am using is: There is an element $x \in E^*(\mathbb{C}P^{\infty})$ such ...
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1answer
68 views

There is given spectrum for $\phi$. Find such $\psi$ with spectrum with special property.

We have spectrum for $\phi$. It is set $X$. Now, we try to find such formula $\psi$ that its spectrum is $Y=\{n+m|n,m\in X\}$. It is allowed to extend sygnature. Being honestly, I can't deal ...
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112 views

Edge homomorphisms for Atiyah-Hirzebruch spectral sequence for a spectrum $X$

I'm interested in some nice identifications (together with explanations) about the edge homomorphisms in the AHSS for a spectrum $X$. $$\times \times \times $$ Let $X$ be a connective spectrum, as ...
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Homotopy fixed points of KU

Let $KU$ be the spectrum representing complex $K$-theory and $KO$ be the spectrum representing real $K$-theory. Complex-conjugation of complex vector bundles lifts to a $C_2$-action on $KU$. It's a ...
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102 views

A problem with a definition of homology of a spectrum

I'm currently reading some notes about the James spectral sequence (here) and there is a passage which is bothering me (page 749 bottom): $$ colim_n h_{p+q}(T(\xi_n))\cong h_{p+q}(M\xi)$$ where $M\...
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1answer
144 views

Is a $\Omega$-Spectrum a connective one?

I can't find this result anywhere, but it seems pretty straightforward. I want to avoid silly mistakes, but I can't see any fault. I'd love to receive some feedback Let $X$ be a $\Omega$-spectrum (of ...
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1answer
110 views

The identification $G=\Omega^\infty \Sigma^\infty S_0$ of the stable group of self homotopy equivalences of spheres with the suspension spectrum

My topology professor told me in a discussion that the suspension spectrum $colim \Omega_n \Sigma_n S_0$ is the same as the monoid $G$ where $G=colim G_n$ where $G_n$ are self homotopy equivalences of ...
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1answer
48 views

Proof of :$H^0(E;\pi_0E)\cong \hom_{\pi_0}(H_0(E;E_0);\pi_0E)$ for $E$ a multiplicative spectrum.

Let $E$ be a multiplicative spectrum, connective, and assume $\pi_0E$ is cyclic. I want to prove that $$H^0(E;\pi_0E)\cong \hom_{\pi_0E}(H_0(E;\pi_0E);\pi_0E)$$ Recall that $H^0(E;\pi_0E):= [E; K(\...
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1answer
29 views

How to show this inequality

Suppose $0=\lambda_0\le\lambda_1\le \ldots \le\lambda_n$ be the eigen values of the normalized laplacian of a graph $G$. Show that $\lambda_1\ge \dfrac{1}{D\text{vol}G}$ where $D$ denotes the ...
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1answer
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Proving that for a multiplicative (B,f)-structure $\mathfrak{B}$ (or X-strcture, or B-structure), Thom spectrum $M\mathfrak{B}$is a ring spectrum.

I'm interested in filling the detail of the claim I made above. I'm following Kochman's notation (page 14 for a def.). Actually he never claims it, (he never spoke about ring spectra), but I think ...
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1answer
48 views

How to prove that $MU$ is an oriented spectrum? A doubt in the proof in Kochman's book

I want to show that the Thom spectrum $MU$ is oriented, namely I want to find a class $x \in \widetilde{MU}^2(\mathbb{C}P^{\infty})$ whose restriction to $\widetilde{MU}^2(S^2)$ is a generator. in his ...
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1answer
103 views

Building $MU$-spectrum via the definition of a $(B,f)$-structure

I want to construct the them spectrum $MU$ using the definition of spectrum associated to a $(B,f)$-structure. Here are the relevant definitions: A $(B,f)$-structure is a collection of strictly ...
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1answer
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Computing the “limit” of a SSeq with $E_{\infty}^{*,*}$ a free graded $\Gamma$-module.

I'm trying to prove the following proposition from Kochman's book. For completion I will write it here the relevant part: Let $E$ be an oriented spectrum with orientation class $x\in E^2(\mathbb{C}...
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Is every infinite loop space a ring prespectrum?

Suppose you are given an infinite loop space $T_0$. The $i$th deloopings, $T_i$ then form a prespectrum with evaluation maps maps $\Sigma T_i \to T_{i+1}$. In the two examples that I know of, the ...
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A doubt concerning axioms - first properties for an homology theory on Spectra

On Rudyak's "On Thom spectra, orientability, and Cobordism", the following axioms are given: Let $\mathscr{S}$ be the category of spectra, and let $\Sigma\colon \mathscr{S}\to \mathscr{S}$ be the ...
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1answer
49 views

Homotopy group of a spectrum $\{ E_n,s_n\}$ as the colimit of $\pi_{k+N}(E_{N})$

I want to prove that the following two definitions of homotopy group of a spectra are equivalent: DEF 1 Let $S= \Sigma^{\infty}S^0$ be the spectrum of spheres. The group $[\Sigma^kS,E]$ is called ...