# 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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1 vote
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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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### 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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### 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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### 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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### $\sigma (xy) \subseteq \sigma (x)\sigma (y)$ 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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### 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
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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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### 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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### 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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### 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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$E$ 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 $\... • 155 3 votes 0 answers 99 views ### Does trivial cohomology of spectra imply trivial homology? It is known that for any spectrum$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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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 ...
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?