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.
91
questions
2
votes
0
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85
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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 $\tilde{I}$, ...
- 33
1
vote
1
answer
64
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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 ...
- 33
2
votes
2
answers
62
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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) ...
- 6,802
0
votes
0
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27
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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 ...
- 1,804
0
votes
1
answer
59
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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 ...
- 41
0
votes
0
answers
53
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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 ...
- 3,045
0
votes
1
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61
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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 ...
- 31
0
votes
1
answer
58
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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)...
- 9,691
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0
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31
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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 ...
- 662
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votes
2
answers
81
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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 $\...
- 7,705
2
votes
1
answer
51
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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 ...
- 662
0
votes
0
answers
23
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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 ...
- 1
0
votes
0
answers
8
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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 ...
- 123
1
vote
0
answers
44
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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 ...
- 723
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42
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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
$f: E \to F$ ...
- 623
3
votes
1
answer
71
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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, ...
- 2,980
1
vote
1
answer
94
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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 ...
- 623
0
votes
1
answer
41
views
$\Omega$-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\\
\{ *\} ...
- 416
8
votes
1
answer
105
views
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 ...
- 416
1
vote
0
answers
43
views
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 ...
- 383
4
votes
0
answers
111
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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 ...
- 121
1
vote
1
answer
67
views
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_{...
- 11
0
votes
0
answers
28
views
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 ...
- 1,376
6
votes
1
answer
126
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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&...
- 393
5
votes
3
answers
457
views
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 ...
- 1,384
3
votes
0
answers
88
views
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 ...
- 393
3
votes
1
answer
154
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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 ...
- 1,384
3
votes
0
answers
168
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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 ...
- 2,089
2
votes
1
answer
78
views
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 ...
- 535
1
vote
0
answers
78
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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, ...
- 2,089
1
vote
1
answer
91
views
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 ...
- 1,253
1
vote
0
answers
47
views
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 ...
- 51
0
votes
1
answer
55
views
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 $$H_*(E;G)=\mathrm{...
- 2,089
1
vote
0
answers
45
views
Symmetric spectrum associated to a ring
In this paper by Dwyer, Greenlees and Iyengar, we are introduced to symmetric spectra, and more particularly to the notion of an $\mathbb{S}$-algebra.
In both the Notation and terminology ($1.5$), ...
- 2,980
2
votes
1
answer
104
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Spectrum of the operator $T \in \mathcal{L}(L^2(\Bbb{R}_+))$ defined by $(Tf)(x)=(1−e^{−x})f(x)$
I'm preparing a mathematical physics exam (the last one) and one of the topic is the spectral theory on operators.
I am totally confused on some concept, so please pardon me.
I am studying this ...
- 470
1
vote
1
answer
140
views
Homology of a spectrum
Let $X$ be a spectrum and $E$ another spectrum (it'll be our coefficients, if it makes things easier I'm ok woth assuming $E=H\mathbb Z$)
The definition of $E_nX$ is usually given as $\pi_n(E\wedge X)...
- 41.4k
2
votes
1
answer
227
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Spectral sequence for homotopy (co)limits
In the accepted answer to this question, user Cary states "What made this spectral sequence tick is that homology/cohomology takes a cofiber sequence to a long exact sequence.".
However this doesn't ...
- 41.4k
0
votes
1
answer
64
views
Is a complex number $\lambda$ in the resolvent set $\rho(T)$ of $T$?
Let $\lambda \notin \{0_{\mathbb C}\} \cup \sigma_p(T)$. Show that $\lambda$ is in the resolvent set $\rho(T)$ of $T \in \mathcal L(\ell^2(\mathbb C))$, where
$$
T(x_n) = \left(\frac{x_n}{n}\right)\,.
...
- 691
3
votes
0
answers
40
views
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 $Y \...
- 1,595
4
votes
0
answers
74
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$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 $E$...
- 127
0
votes
0
answers
32
views
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}^{...
- 11
1
vote
1
answer
50
views
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 ...
user720939
2
votes
1
answer
75
views
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 ...
- 41.4k
2
votes
0
answers
81
views
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, ...
- 127
2
votes
0
answers
165
views
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 ...
- 2,185
1
vote
0
answers
95
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 ...
2
votes
1
answer
49
views
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 $\...
- 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 ...
- 821
2
votes
0
answers
91
views
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 ...
- 594
0
votes
0
answers
23
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?
