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Mathematics Stack Exchange

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.

102 questions
11
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Is there a "nice" description of Bousfield localisation at $H \mathbb{Z}$?

Let $\newcommand{\Sp}{\mathrm{Sp}}\Sp$ denote the $\infty$-category of spectra and $\newcommand{\Z}{\mathbb{Z}} H\Z$ the Eilenberg-Mac Lane spectrum of the integers. Given any spectrum $X \in \Sp$, I am interested in understanding its Bousfield…
Ben Steffan
  • 8,325
9
votes
2 answers

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 maps $\Sigma X_{n} \rightarrow X_{n+1}$ for all $n…
Sunny Sood
  • 866
8
votes
1 answer

Are spectra representing cohomology theories unique?

I know that every generalised (Eilenberg-Steenrod) cohomology theory defines a spectrum (in the sense of Lewis-May), and vice-versa. I also know that maps between spectra are richer than maps between cohomology theories (due to the existence of…
user09127
  • 451
8
votes
1 answer

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 $x\in E^2(\mathbb{C}P^{infty})$, then $$ E^*(\mathbb{C}P^n)\cong \pi_*E…
Riccardo
  • 7,583
7
votes
0 answers

"Naive" smash products for spectra

Suppose I work in the completeley naive homotopy category of spectra, by which I mean sequences $E = (E_n)_{n = 0, 1, \dots}$ together with maps $\sigma_{E,n}: S^1 \wedge E_n \to E_{n+1}.$ We might require the $E_n$ to be CW complexes and the…
Tom Bachmann
  • 1,156
6
votes
2 answers

String/surface diagrams and the sphere spectrum

I am trying to understand intuitively how the spectrum $\mathbb{S}$ of the stable homotopy groups of spheres arises from the diagrammatic approach hinted at e.g. in John Baez's TWF 102, by explicitly finding the first few stable homotopy groups…
pregunton
  • 6,358
6
votes
0 answers

Nullhomotopicity of $\mathbb{S}/p \to^p \mathbb{S}/p$ for $p=2$ and $p \neq 2$?

For a given spectra $X$ we have $X/p$ defined as the cofiber $X \to^p X$ where the map is basically defined via defining it on the sphere spectrum $\mathbb{S}$! To define $\cdot p$ on the sphere spectrum we just need to define it on $\mathbb{S^1}$…
user135743
  • 439
6
votes
2 answers

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 $\Sigma X_i \rightarrow X_{i+1}$ given by…
Jonas Linssen
  • 12,065
6
votes
1 answer

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" Lemma 6.1. The following equalities hold in…
Pierre Elis
  • 393
6
votes
1 answer

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 homotopy theory and in particular about spectra. I…
Sofie Verbeek
  • 1,149
6
votes
2 answers

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 that it's restriction to $\mathbb{C}P^1$ is a…
happymath
  • 6,297
5
votes
0 answers

Complex of sheaves, Eilenberg-MacLane spectra and hypercohomology

This question is about the relation between the category of spectra and the category of chain complexes of abelian groups. Specifically, I am trying to understand the examples from Deligne cohomology. The Deligne complex $\mathbb{Z}(n)$ is defined…
timaeus
  • 331
5
votes
0 answers

$KO_*$ 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 $E$ to be complex orientable. Naturally, I looked up…
Excalibur
  • 147
5
votes
0 answers

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 $[HF _ p \wedge X , \sum^i HF _ p] _{ HF_p−module}=0$ for all i. So, we get…
Surojit
  • 945
5
votes
1 answer

How is $p$-localization of spectra defined?

I have a question regarding the construction of the $p$-localization of a given spectrum $X$. I have seen in many papers people defining the dual notion of it, namely the $p$-completion of $X$, $X_p^{\wedge}:= L_{M(\mathbf{Z}/p)}X$, but never…
user430191
  • 781
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