1
vote
0answers
102 views

Existence of open dense subset in a Lie group

Let $G=S\cdot R$ be a Levi-Malcev decomposition of a connected complex Lie group and $\Gamma$ a discrete subgroup of $G$ such that the subgroups $\Gamma_g := \Gamma\cap (gSg^{-1})$ are finite for all ...
7
votes
1answer
301 views

What are the applications of Dowker's theorem?

Let $R \subset X \times Y$ be any relation between sets $X$ and $Y$. CH Dowker constructed two simplicial complexes $K$ and $L$ associated to $R$: a simplex in $K$ consists of finitely many elements ...
6
votes
2answers
520 views

Computer aided homology computations

Background I am currently working on the homology of some modulispace and there exists a much simpler chaincomplex with the same homology. It is a quotient of a bisimplicial complex by a subcomplex. ...
10
votes
0answers
448 views

Codimension Two Embeddings in Goodwillie-Weiss Manifold Calculus, and the Difficulty of Fundamental Groups

In manifold calculus, there are various analyticity estimates which run into trouble for codimension two embeddings. For instance, the functor $\operatorname{Emb}(M,N)$ is analytic in $M$ if $\dim M ...
3
votes
1answer
384 views

Definition of Pontrjagin Classes

I am studying characteristic classes recently and find some quesions about Pontrjagin classes. Firstly the definition of Pontrjagin classes is not that natural. When we talk about Pontrjagin classes ...
9
votes
4answers
892 views

Coincidences amongst classifying spaces and Eilenberg Mac-Lane spaces

Given that $$\mathbb{R}P^{\infty} = B O(1) = K(\widehat{O(1)}, 1)$$ $$\mathbb{C} P^{\infty} = B U(1) = K( \widehat{U(1)}, 2)$$ is there any way to make sense of $$\mathbb{H}P^{\infty} = B Sp(1)$$ in a ...
3
votes
2answers
1k views

How does singular homology H_n capture the number of n-dimensional “holes” in a space?

This is a foundational doubt I have. How does singular homology H_n capture the number of n-dimensional holes in a space? We disregard the case of $H_0$ as it has the very satisfactory explanation ...
3
votes
2answers
426 views

Euler characteristics and operator indices as exponents for Laurent polynomials

This question is rather vague. Are there any natural situations which involve Laurent polynomials of the form $$\sum q^{a_i}\in\mathbb{Z}[q,q^{-1}]$$ where the $a_i$'s are either Euler characteristics ...
5
votes
2answers
513 views

visualizing what's going on in based homotopy theory, et al.

I'm reading J.P. May's Concise Course in Algebraic Topology, and I'm having a lot of trouble visualizing how things work in Chapter 8, "Based cofiber and fiber sequences". Of course this is pretty ...
19
votes
4answers
2k views

Visualizing how Cech cohomology detects holes

I think it's pretty intuitive how singular/simplicial cohomology detects "holes" in a space. How can we directly visualize how and in what sense the Cech cohomology of a cover does this? In ...