Questions tagged [spectra]

Use this tag for questions 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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Confusions about function spectrum in Anderson duality

In appendix B of this paper https://arxiv.org/abs/math/0211216, Hopkins and Singer defined the Anderson dual $\tilde{I}(E)$ of a spectrum $E$ as the function spectrum of maps from $E$ to $\tilde{I}$, ...
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Why is $X \mapsto hom(\pi_*^{st}X, \mathbb{Q})$ the same as ordinary rational cohomology?

I am trying to understand the notion of Anderson duality from appendix B of this paper https://arxiv.org/abs/math/0211216 by Hopkins and Singer. But I somehow get stuck at the very first steps. I am a ...
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Loop-Suspension adjunction unit is stable equivalence

Let $X$ be a sequential spectrum, then the unit of the $(\Sigma,\Omega)$-adjunction yields a map $\eta : X \to \Omega \Sigma X$. The authors of the book Foundations of Stable Homotopy Theory (p. 48) ...
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($\infty$ ,1) construction of Spectra and stabilization

I am trying to understand Spectra as an attempt to make the infinity category of pointed spaces stable. I don't know the details of some $\infty$,1 constructions so I have several questions. From now ...
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Is there is any relationship between determinant and spectral radius of the matrix?

We all know that determinant is the product of the eigen values of a matrix. I have found some general term for the determinant of the adjacency matrices from some series of graphs. Can i establish ...
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Relation between reduced and unreduced cohomology

I have found a lemma in nlab I can't quite make sense of (http://nlab-pages.s3.us-east-2.amazonaws.com/nlab/show/generalized+(Eilenberg-Steenrod)+cohomology#FromUnreducedToReducedCohomology). It ...
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Spectral value of reduced von Neumann algebra

Let $M$ a von Neumann algebra and $x$ positive and $a$ a projector, $N_x$ is the von Neumann algebra generated by $x$. Do you know if $min Spec_{aN_xa}(axa)\in Spec_M(x)$ please? [Attempt added from ...
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Definition of a map of (pre)spectra in HoTT

I'm trying to pick up some homotopy theory by reading various HoTT sources (I find it much easier than reading classical homotopy theory books). In Floris Van Doorn's PhD thesis he defines (pre)...
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$\mathbf H R$ is a ring spectrum if $R$ is a ring

The first example of a ring spectrum is probably the Eilenberg-McLane spectrum of a ring $R$. But how is the multiplication $\mu: \mathbf H R \wedge \mathbf H R \to \mathbf H R$ defined? Probably this ...
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What is the essential image of the suspension spectrum functor $\Sigma^\infty$?

Let $\mathsf{hCW}$ denote the homotopy category of CW-complexes and $\mathsf{hCWSpec}$ the homotopy category of CW-spectra (ie. families of CW-complexes $(X_i)_{i\in\mathbb{Z}}$ with connection maps $\...
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Excision in the stable homotopy category

Is there a way to make precise the statements (if it holds at all) that excision holds in the stable homotopy category? I am a beginner in this kind of things, and the stable homotopy category for me ...
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Magnitude Spectrum from real 2D Fourier Coefficients

I know how to calculate the coefficients for a 2D Fourier Series as shown: 2D real Fourier Series I calculated the Coefficients A[m][n], B[m][n], C[m][n] and D[m][n]. Now I want to create a magnitude ...
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Generalize Power spectrum density to higher orders

If the power spectrum density for a given time series is given by : $PSD(f) = \frac{2 \Delta}{N} \sum_{j=1}^{N} \delta B^{2}(t_{j}, f)$, where, where $δB(t_{j} , f)$ is the magnitude of the trace ...
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Showing that $\sigma_A (a) = \sigma_C (A).$

Let $A$ be a unital $C^*$-algebra and $C$ be a $C^*$-subalgebra containing the identity of $A,$ and let $a$ be a self-adjoint element in $C.$ Then $\sigma_A (a) = \sigma_C (a).$ It's quite clear that ...
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stable homotopy category Ho(Spectra) of category of spectra

The category of spectra over CW-complexes has as objects sequences $E:= \{E_n \}_{n \in N}$ of CW complexes $E_i$ together with structure maps $S^1 \wedge E_n \to E_{n+1}$. The morphisms $f: E \to F$ ...
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Notation clarification in stable homotopy theory

In this paper which surveys the Goodwillie Calculus, the following notation is used in the introduction: $$\pi_*(P_1 I(X)) \cong \pi_*^s(X)$$ where $X$ is a based space, $I$ is the identity functor, ...
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Discrete Spectra

What is a "discrete spectra" in context of homotopy theory/ derived category theory? It is for example mentioned here. Although it looks quite "googleable" I found nowhere a ...
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$\Omega$-spectrum from homotopy-cartesian squares?

Let $\{X_{n}\}_{n\geq0}$ be a sequence of (pointed) spaces such that for all $n \geq 0$, we have homotopy-cartesian squares $\require{AMScd}$ \begin{CD} X_{n} @>>> \{* \}\\ @VVV @VVV\\ \{ *\} ...
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8 votes
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Visualising Spectra?

I have recently started to learn about Spectra. To state the definition (that I have learned), a spectrum $X = \{X_{n} \}_{n \geq0}$ is a sequence of based spaces $X_{n}$, with basepoint preserving ...
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Checking $X \to S[J^{-1}] \wedge X$ is an $S[J^{-1}]$ equivalence.

This is surely a trivial question, since all sources seem to relegate it to an exercise. Here's my attempt at a proof. Is this the idea? Please excuse the wordiness of the exposition, it's from my own ...
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4 votes
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Spectra smashing with $\mathbb{S}^1$ is equivalent to shifting

I'm struggling with understanding the proof that the suspension isomorphism in the homotopy category of topological spectra is equivalent to smashing with the sphere $\mathbb{S}^1$. The proof I'm ...
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Commutativity of naive smash product

I am reading Switzer right now, and I came upon this line in the definition of naive smash product of spectra (p.256): Some of the naive smash products are commutative ($E\wedge_{BC}F \simeq F\wedge_{...
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How to analyze the differences between experimental and theoretical data.

I have around 5000 data points of theoretically computed and experimentally measured (via FTIR) K2 line positions(v (v", J''; v', J'). I divided the data into wn segments, so that I could analyze ...
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6 votes
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Basic computation in Steenrod algebra

In Homology operations for $H_\infty$ and $H_n$ spectra (pdf), Steinberger makes the computation of the Dyer-Lashof operations in $H\mathbb F_p$, and at some point uses the following "basic fact&...
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5 votes
3 answers
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Show $T$ is compact operator if $\langle Te_n,e_n \rangle$ tend to zero.

Suppose $\mathcal{H}$ is a Hilbert space, and $T\in B(\mathcal{H})$. If for each orthonormal (norm 1) basis $\{e_n\}\subseteq \mathcal{H}$, we have $\langle Te_n, e_n \rangle \rightarrow 0$. Can we ...
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Which (co)homology theories have (co)chain complexes, spectrum edition?

What homological functors $\mathbf{Sp}\rightarrow\mathbf{Ab}$ arise in the form $H_0\circ T$, for $T:\mathbf{Sp}\rightarrow D(\mathbf{Ab})$ a triangulated functor, and dually for cohomological ...
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1 answer
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$\sigma (xy) \subseteq \sigma (x)\sigma (y)$ in Banach algebra.

Let $A$ be a Banach algebra(with unit $e$), $x,y \in A$ and $xy=yx$. Prove that $\sigma(xy) \subseteq \sigma (x) \sigma (y)$, also $\sigma (x+y)\subseteq \sigma(x)+\sigma(y)$. Where $\sigma$ means the ...
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Mahowald-Hopkins theorem

I am trying to recover fully a proof of a result which is attributed to M. Hopkins in the litterature, namely the following: Let $h:S^1\to BGL_1(\mathbb S_p)$ detect the element $1-p$, then the Thom ...
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2 votes
1 answer
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Why is the Moore spectrum $S\mathbf{Z}_{(J)}$ a ring spectrum?

Let $J$ be a set of primes and consider the Moore spectrum $S\mathbf{Z}_{(J)}$. In his paper 'The localization of spectra with respect to homology', Bousfield writes that $S\mathbf{Z}_{(J)}$ is a ring ...
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A non modern theory of generalized Thom spectra

I've already posted this question on mathoverflow a few days ago and I had no reaction, I hope it is not illegal to repost it on math.stackexchange. I'm new in the subject of stable homotopy theory, ...
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1 vote
1 answer
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Equivalence between lax monoidal functors and monoids in the functor category.

I'm trying to go through the details of Proposition 3.4 of: https://ncatlab.org/nlab/show/Day+convolution For whatever reason, I don't see how to translate the conditions of a lax monoidal functor ...
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1 vote
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The infinite loop space associated to the spectrum associated to a special $\Gamma$-space

I am currently reading M. Mandel's paper `An Inverse $K$-theory Functor' to extract some results I may wish to use in my masters research project, and I am stuck on one particular claim. On the top of ...
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Basic question about the definition of the homology of a spectrum

The general definition of the homology of a spectrum $E$ with coefficients in an abelian group $G$ is $$H_*(E;G):=\pi_*(E\wedge HG)$$ and I always see people using the equality $$H_*(E;G)=\mathrm{...
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1 vote
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Symmetric spectrum associated to a ring

In this paper by Dwyer, Greenlees and Iyengar, we are introduced to symmetric spectra, and more particularly to the notion of an $\mathbb{S}$-algebra. In both the Notation and terminology ($1.5$), ...
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2 votes
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Spectrum of the operator $T \in \mathcal{L}(L^2(\Bbb{R}_+))$ defined by $(Tf)(x)=(1−e^{−x})f(x)$

I'm preparing a mathematical physics exam (the last one) and one of the topic is the spectral theory on operators. I am totally confused on some concept, so please pardon me. I am studying this ...
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1 answer
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Homology of a spectrum

Let $X$ be a spectrum and $E$ another spectrum (it'll be our coefficients, if it makes things easier I'm ok woth assuming $E=H\mathbb Z$) The definition of $E_nX$ is usually given as $\pi_n(E\wedge X)...
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Spectral sequence for homotopy (co)limits

In the accepted answer to this question, user Cary states "What made this spectral sequence tick is that homology/cohomology takes a cofiber sequence to a long exact sequence.". However this doesn't ...
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Is a complex number $\lambda$ in the resolvent set $\rho(T)$ of $T$?

Let $\lambda \notin \{0_{\mathbb C}\} \cup \sigma_p(T)$. Show that $\lambda$ is in the resolvent set $\rho(T)$ of $T \in \mathcal L(\ell^2(\mathbb C))$, where $$ T(x_n) = \left(\frac{x_n}{n}\right)\,. ...
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A homotopy equivalence of $X \wedge Y \to Y \wedge X$ to itself.

In the Adams Bluebook, to define a smash product of CW-spectra, the author uses a homotopy equivalence between two smash product of compactly generated pointed spaces from $S^{1} \wedge Y$ to $Y \...
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4 votes
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$KO_*$ groups of $\mathbb{R}P^\infty$, "Snaiths" theorem for $KO$

I wonder whether anyone has taken the time to compute $KO_*(\mathbb{R}P^\infty)$? The standard tools to compute these Groups in the complex case rest on the requirement for the cohomology theories $E$...
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Spectral properties via determinant function

I searched through literature but could not find any related topic for the question below. I hope some of you may be able to point me to the right direction. Let $X: \mathbb{R}\rightarrow\mathbb{R}^{...
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1 answer
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Field spectra and Eilenberg--MacLane spectra?

Apparently thanks to a theorem of Hopkins and Smith, every field spectrum splits into a wedge of Morava K-theories, where we allow the cases $K(0) = H \mathbb{Q}$ and $K(\infty) = H \mathbb{F}_p$. I ...
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2 votes
1 answer
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Why is $E^*(X)$ graded commutative?

Context: I'm reading The stable homotopy category by Malkiewich (see his webpage, at Expository writings), and at this point the author hasn't actually introduced spectra, just the properties we would ...
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2 votes
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Functoriality of twisted K-theory

In order to avoid the XY problem I will first state what I want, then what I think is the solution and how that failed until now. I'm trying to use May-Sigurdsson's definition of Twisted $K$-theory, ...
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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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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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2 votes
1 answer
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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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3 votes
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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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Spectral decomposition of an operator

Given the operator: $$A=\begin{pmatrix} i & 0 & - 4\\ 0 & - 3i & 0\\ 2 & 0 & - i \end{pmatrix}$$ Now $\det(zI-A)=(z+3i)(z^2+9)=(z+3i)^2(z-3i)$, I know a theorem that says that ...
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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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