I'm reading Carlsson's "A survey of equivariant homotopy theory" and I have a question. Let G be a topological group and X be a G-space (for a nice notion of "space"). He defines XhG=EG*GX the homotopy orbit space, and XhG=F(EG,X)G the homotopy fixed point space. He claims there are spectral sequences E2p,q=Hp(G;Hq(X))=>Hp+q(XhG) and Ep,q2=H−p(G;πq(X))=>πp+q(XhG). Now, if I'm not mistaken the first one follows from the Serre spectral sequence applied to the Borel fibration X->XhG->BG (take the fiber bundle with fiber X associated to the action of G on X and to the G-principal bundle EG->BG). But where does the second one come from?

reinzogoq 2022-09-14 Answered
I'm reading Carlsson's "A survey of equivariant homotopy theory" and I have a question.
Let G be a topological group and X be a G-space (for a nice notion of "space"). He defines
X h G = E G × G X
the homotopy orbit space, and
X h G = F ( E G , X ) G
the homotopy fixed point space.
He claims there are spectral sequences
E p , q 2 = H p ( G ; H q ( X ) ) H p + q ( X h G )
and
E 2 p , q = H p ( G ; π q ( X ) ) π p + q ( X h G ) .
Now, if I'm not mistaken the first one follows from the Serre spectral sequence applied to the Borel fibration X X h G B G (take the fiber bundle with fiber X associated to the action of G on X and to the G-principal bundle E G B G).
But where does the second one come from?
You can still ask an expert for help

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

Solve your problem for the price of one coffee

  • Available 24/7
  • Math expert for every subject
  • Pay only if we can solve it
Ask Question

Answers (1)

yamalwg
Answered 2022-09-15 Author has 17 answers
It should be a special case of the Bousfield-Kan spectral sequence for homotopy limits. You can think of it as a "Grothendieck spectral sequence" associated to the "derived functors" of taking fixed points and taking π0 (which are, respectively, taking homotopy fixed points / group cohomology and taking homotopy groups).

We have step-by-step solutions for your answer!

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

You might be interested in

asked 2022-09-11
A concert venue is located 4 minutes from a large university. A group of students are planning to attend the concert and think they can get there in less time. They test this hypothesis by using social media to survey a sample of other drivers to see how long this trip took them. The null and alternative hypotheses are given.
....
Here is the full question:
(a) what is the consequence of a Type I error in the context of this problem?
hey conclude it will average rewer than 4 minutes to get there, but the real average is more than 4 minutes.
They conclude that it will average more than 4 minutes, but the real average is 4 minutes to get there.
They conclude it will average 4 minutes to get there, but the real average is less than 4 minutes.
They conclude it will average rewer than 4 minutes, but the real average is 4 minutes to get there.
(6) What are the consequences of making a Type II error in the context of this problem?
They conclude it will average fewer than 4 minutes to get there, but the real average is more than 4 minutes.
They conclude that it will average more than 4 minutes, but the real average is 4 minutes to get there.
They conclude it will average 4 minutes to get there, but the real average is less than 4 minutes.
They conclude it will average fewer than 4 minutes, but the real average is 4 minutes to get there.
asked 2022-09-24
Statement:
A manufacturer has been selling 1000 television sets a week at $480 each. A market survey indicates that for each $11 rebate offered to a buyer, the number of sets sold will increase by 110 per week.
Questions :
a) Find the function representing the revenue R(x), where x is the number of $11 rebates offered.
For this, I got (110x+1000)(480−11x). Which is marked correct
b) How large rebate should the company offer to a buyer, in order to maximize its revenue?
For this I got 17.27. Which was incorrect. I then tried 840999, the sum total revenue at optimized levels
c) If it costs the manufacturer $160 for each television set sold and there is a fixed cost of $80000, how should the manufacturer set the size of the rebate to maximize its profit?
asked 2022-09-07
Rings for which the Köthe conjecture holds
Is there an overview of rings for which the Köthe conjecture is known to hold? In particular, I am interested in endomorphism rings of graded modules over multivariate polynomial rings. I read one survey states that the conjecture holds for left artinian rings (alas, without a reference). However, in the notion of that survey, to hold means that "the ideal generated by every nil left ideal of R is nil."
So my explicit question is: In any left artinian ring, or in a particular special case thereof, is it true that the sum of two left nil ideals is nil again?
asked 2022-09-11
A question in GRE states:
In a survey of a town,it was found that 65% of the people surveyed watched the news on television,40% read newspaper, and 25% read a newspaper and watched the news on television. What percent of the people surveyed neither watched the news on television nor read a newspaper.
asked 2022-09-03
Ice cream flavorsA survey was taken among 600 middle school students for their preferred ice cream flavours.- 250 like strawberry- 100 liked both strawberry and vanilla- 130 like strawberry but not chocolate- 120 like only vanilla- 300 like only one flavor- 270 students said they liked vanilla- 30 students like all three flavorsQuestions:- How many do not like any of the 3 flavors?- How many like at least one of the 3 flavors?- How many like at least 2 flavors?- How many like chocolate?
asked 2022-09-11
A certain town has 25,000 families. These families own 1.6 cars, on the average; the SD is 0.90. And 10% of them have no cars at all. As part of an opinion survey, a simple random sample of 1,500 families is chosen. What is the chance that between 9% and 11% of the sample families will not own cars?
So I'm looking at the solution of this question it shows that to get the Standard Error of the Sum is SD= 0.9   0.1 However, I thought the formula to get SE is:
SE= SD of the Box N u m b e r o f D r a w s
In this case, isn't the Box of the SD given at 0.9? Why do we still use the bootstrap method to get SD= 0.9   0.1 =0.3 ??
Many thanks in advance!
asked 2022-09-25
I had this question when I was shopping for some rates to refinance my house: how do know if I have found the lowest rates after searching through a bunch of offers with a reasonable amount of confidence, in a mathematical/statistical term. For example, if found "x" offers from different vendors, is the amount of "x" large enough to represent the bigger population i.e. all offers out there?
The reason I am asking is that I vaguely remember there are statistical sampling techniques from college days that survey companies used to represent the bigger populations. For instance, after surveying 2000 people, they can conclude something for the U.S. population. In other words, they "know" 2000 people can represent the U.S. population with some scientific backing.
Just curious, if there is a way to figure out the magical number "x", then I can go with the lowest rate after browsing through "x" vendors as well. The question is purely theoretical in this simple case: the lowest rate wins. Any leads or ideas are appreciated.

New questions