All Questions
8,725 questions
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free homotopy groups -- when do they exist?
Let (X,x) be a pointed space. There is an action of π1(X,x) on πn(X,x) -- determined by considering πn(X,x)=πn-1(ΩxX,x), where ΩxX denotes the space of loops in X based at x, ...
- 6,069
7
votes
1
answer
282
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Can you construct a mapping space from local data? (looking for reference)
I'd to know if/where there is a reference for the following construction.
Let C_*(maps(M, T)) denote the singular chains on the space of continuous maps from an n-...
- 12.8k
30
votes
3
answers
3k
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Why is homology not (co)representable?
This is in the same vein as my previous question on the representability of the cohomology ring. Why are the homology groups not corepresentable in the homotopy category of spaces?
- 5,548
9
votes
3
answers
1k
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Representablity of Cohomology Ring
I know that the individual cohomology groups are representable in the homotopy category of spaces by the Eilenberg-MacLane spaces. Is it also true that the entire cohomology ring is representable? If ...
- 5,548
4
votes
2
answers
2k
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proving that an inclusion map from a subcomplex is a homotopy equivalence
This is a pretty basic question but I have been stuck on it for a while.
Given an abstract simplicial complex X and a subcomplex A, why does * suffice to show that the map |A|->|X| induced by ...
- 13.6k
7
votes
1
answer
597
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Pontryagin product from an operad
For a topological group G, we have a Pontryagin product in homology by multiplying representative cycles. This gives the homology the structure of an associative graded algebra. Am I correct in ...
- 71
3
votes
3
answers
2k
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singular cohomology of SO(4)
I'm trying to compute the singular cohomology of SO(4), just as practice for using spectral sequences. I got H0=Z, H1=0, H2=Z/2Z, H3=Z⊕Z, H4=0, H5=Z/2Z, and H6=Z. Are these correct? I'm not ...
- 6,069
22
votes
13
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8k
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Category theory sans (much) motivation?
So I have a friend (no, really) who's taking algebra and is struggling to gain intuition for it. My story is as follows: I used to hate abstract algebra, with pretty much a burning passion, until I ...
17
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8
answers
3k
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Smooth classifying spaces?
Take G to be a group. I care about discrete groups, but the answer in general would be welcome too. There are the various ways to construct the classifying space of G, bar construction, cellular ...
- 1,422
14
votes
1
answer
896
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Commutativity in K-theory and cohomology
The Chern classes give a map $f : BU \to \prod_n K(\mathbb{Z},2n)$, which is a rational equivalence. However, it is not an equivalence over $\mathbb{Z}$ because the cohomology of $BU$ is just a ...
- 31.2k
21
votes
5
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How to compute the (co)homology of orbit spaces (when the action is not free)?
Suppose a compact Lie group G acts on a compact manifold Q in a not necessarily free manner. Is there any general method to gain information about the quotient Q/G (a stratified space)? For example, I ...
- 211
53
votes
8
answers
9k
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Analogue to covering space for higher homotopy groups?
The connection between the fundamental group and covering spaces is quite fundamental. Is there any analogue for higher homotopy groups? It doesn't make sense to me that one could make a branched ...
- 13.6k
80
votes
7
answers
20k
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Teaching statements for math jobs?
What is the purpose of the "teaching statement" or "statement of teaching philosophy" when applying for jobs, specifically math postdocs? I am applying for jobs, and I need to write one of these ...
22
votes
5
answers
7k
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Describing the universal covering map for the twice punctured complex plane
As is well known, the universal covering space of the punctured complex plane is the complex plane itself, and the cover is given by the exponential map.
In a sense, this shows that the logarithm has ...
- 5,530
24
votes
3
answers
5k
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Euler characteristic of a manifold and self-intersection
This is probably quite easy, but how do you show that the Euler characteristic of a manifold M (defined for example as the alternating sum of the dimensions of integral cohomology groups) is equal to ...
- 5,530
41
votes
7
answers
5k
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Simplicial objects
How should one think about simplicial objects in a category versus actual objects in that category? For example, both for intuition and for practical purposes, what's the difference between a [...
- 21k
8
votes
2
answers
1k
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Differentials in the Lyndon-Hochschild spectral sequence
The Lyndon-Hochschild(-Serre) spectral sequence applies to group extensions in a manner analogous to the Serre-Leray spectral sequence applied to a fibration.
Does anyone know of a good description (...
- 1,422
76
votes
9
answers
15k
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understanding Steenrod squares
There is a function on $\mathbb{Z}/2\mathbb{Z}$-cohomology called Steenrod squaring: $Sq^i:H^k(X,\mathbb{Z}/2\mathbb{Z}) \to H^{k+i}(X,\mathbb{Z}/2\mathbb{Z})$. (Coefficient group suppressed from ...
- 6,069
66
votes
5
answers
8k
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Does homology have a coproduct?
Standard algebraic topology defines the cup product which defines a ring structure on the cohomology of a topological space. This ring structure arises because cohomology is a contravariant functor ...
- 661
104
votes
10
answers
24k
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Motivation for algebraic K-theory?
I'm looking for a big-picture treatment of algebraic K-theory and why it's important. I've seen various abstract definitions (Quillen's plus and Q constructions, some spectral constructions like ...
11
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2
answers
1k
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Elliptic curve over spectra?
Filling the gaps in my knowledge to understand the tmf question.
So, what is the analogue of elliptic curve over the category of spectra?
- 15.1k
44
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7
answers
22k
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How do you show that $S^{\infty}$ is contractible?
Here I mean the version with all but finitely many components zero.
- 10.5k
11
votes
5
answers
3k
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Ribbon graph decomposition of the moduli space of curves
What is a ribbon graph? What is the ribbon graph decomposition of the moduli space of curves? What are some good references for this material?
- 21k
43
votes
5
answers
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How can you tell if a space is homotopy equivalent to a manifold?
Is there some criterion for whether a space has the homotopy type of a closed manifold (smooth or topological)? Poincare duality is an obvious necessary condition, but it's almost certainly not ...
- 31.2k
0
votes
0
answers
2k
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Ignore this question [closed]
This question is a hacky way to create some tags for you to use. Move along.
- 24.1k