# 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.

51 questions
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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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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