All Questions
Tagged with duality at.algebraic-topology
21 questions
5
votes
1
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249
views
Double hom with $\mathbb{CP}^\infty$
Pontrjagin duality gives a double dual theorem for "hom with $S^1$", and $S^1$ is $\textbf{B}\mathbb{Z}$ up to homotopy.
$\textbf{B} \textbf{B}\mathbb{Z}$, modeled by $\mathbb{C}\mathbb{P}^{\...
user30211
2
votes
0
answers
147
views
Explicit S-duality map
$\DeclareMathOperator{\Th}{Th}$
The Thom space of a closed manifold $M$ ($\Th(M)$) is the $S$-dual to $M_+ (=M \cup \{pt\})$. Let $M$ be embedded in $\mathbf R^{n+k}$. I found the duality map from $S^...
1
vote
1
answer
214
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Homology of complements and homotopy equivalence
Albrecht Dold gave a short proof of the Jordan-Alexander complement theorem in the following form.
Given two closed sets $A$, $B \subset {\bf R}^n$ that are homeomorphic, the singular homology groups $...
- 18.7k
8
votes
1
answer
929
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On the Euler characteristic of a Poincaré duality space
Background. Suppose that $M$ is an oriented, connected, closed manifold of dimension $d$ with fundamental class $\mu \in H_d(M;\Bbb Z)$. Let $\Delta : M \to M \times M$ be the diagonal map. Then the ...
- 18.9k
2
votes
0
answers
55
views
Tangential normal invariant isomorphism
Recently, I was reading the paper "Finite Group Actions on Kervaire Manifold" by Crowley, Hambolton. But I am having problem understanding a definition. Here it is,
In page 15-16 they are ...
10
votes
1
answer
403
views
Finite domination and Poincaré duality spaces
Here are some definitions:
A space is homotopy finite if it is homotopy equivalent to a finite CW complex.
A space finitely dominated if it is a retract of a homotopy finite space.
A space $X$ is a ...
- 18.9k
14
votes
1
answer
1k
views
Are Spanier-Whitehead duals of general spaces expressible through some generalization of normal bundles?
The question is inspired by an answer to The concept of Duality
It is explained in that answer that the Spanier-Whitehead dual of a compact manifold is given by the Thom spectrum of normal bundles of ...
- 18.4k
4
votes
0
answers
180
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Spanier-Whitehead dual of space of natural transformations
Let $F, G: \mathcal{J} \to \mathsf{Sp}$ be continuous functors between $\sf{Sp}$-enriched categories, where $\sf{Sp}$ denotes any of the point-set models for spectra (i.e., orthogonal spectra).
...
8
votes
2
answers
430
views
Comparison: Formal Wirthmüller isomorphism of Fausk-Hu-May vs. Balmer et. al
$\newcommand{\Cc}{\mathcal{C}}$
$\newcommand{\Dd}{\mathcal{D}}$
$\newcommand{\tensor}{\otimes}$
$\DeclareMathOperator{\Sp}{Sp}$
This question is about comparing the approaches for a formal Wirthmüller ...
- 556
49
votes
4
answers
4k
views
Why is there a duality between spaces and commutative algebras?
1) The category of affine varieties over $\mathbb{C}$ is equivalent to the opposite category of finitely generated reduced algebras over $\mathbb{C}$. The equivalence associates to an affine variety ...
- 9,491
13
votes
3
answers
607
views
GOE/GSE duality and Bott periodicity
Many papers in random matrix theory make passing references to duality between eigenvalue statistics of the GOE and GSE, for which the most concrete reference I can find is https://arxiv.org/pdf/math-...
- 568
13
votes
2
answers
1k
views
Is there any relationship between the topologies of the clique complex and the independence complex?
Let $G$ be a simple graph on a finite vertex set. The clique complex $X(G)$ is the simplicial complex whose faces are complete subgraphs of $G$, and the independence complex $I(G)$ is the simplicial ...
- 7,857
5
votes
1
answer
349
views
A generalization of integral Poincaré duality
In this paper, Felix, Halperin and Thomas define the notion of a Gorenstein space over a field $\mathbb{k}$:
An augmented differential graded algebra $R$ over $\mathbb{k}$ is Gorenstein if $\text{Ext}...
- 208
35
votes
1
answer
2k
views
Are there topological versions of the idea of divisor?
I am trying to extract a particular, more lightweight and more focussed at the same time, case of my recent question Which of the physics dualities are closest in essence to the Spanier-Whitehead ...
- 18.4k
2
votes
1
answer
602
views
Pushforward in Compactly Supported Cohomology
Suppose $X,Y$ are locally compact Hausdorff spaces and $f:X\to Y$ is a topological submersion of relative dimension $n$. By this we mean that for all points $x\in X$, there exists an open neighborhood ...
- 1,761
2
votes
0
answers
166
views
Do Poincaré duality algebras need to be defined over a field?
I asked the below question here on MSE, but after some time and a bounty offering I have not received an answer.
A graded commutative, connected $\mathbb{k}$-algebra $A$ is called a Poincaré duality ...
- 208
3
votes
0
answers
166
views
Naturality of Poincaré–Lefschetz
Let $X$ be compact and Hausdorff, $A\subseteq B\subseteq X$ both closed such that $X\setminus A$ is an open orientable $d$-manifold. Then also $X\setminus B$ is an open orientable $d$-manifold. We ...
- 1,623
5
votes
0
answers
185
views
Dual Steenrod squares
Fix the ground ring $\mathbb{F}_2$ and let $X$ be a space with finite homology. Then we have an isomorphism $\Phi^i_X:H_i(X)\to H^i(X)^*,a\mapsto \langle-,a\rangle$ which allows us to define the dual ...
- 1,623
7
votes
5
answers
624
views
Dualizable classifying spaces
If $G$ is a finitely generated free group, then its classifying space $B G$ can be presented as a finite CW complex (a finite bouquet of circles), and therefore is Spanier-Whitehead dualizable. Are ...
- 66.8k
5
votes
2
answers
712
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Alexander duality theorem for CW-complexes and stable homotopy theory
In Adams, J.F. Infinite Loop Spaces Princ. Univ. Press. page 9 he states Alexander duality theorem
Theorem:[Alexander Duality] $$ H^r(X,G)=H_{n-r+1}(S^n-X,G)$$
for finite CW-complexes with a "nice ...
- 2,871
6
votes
0
answers
133
views
The metric gives the optimal element in a class
In geometry there is plenty of examples in which the following happens:
Some elements are considered equivalent, in some topological or algebraic sense
We take the quotient
The metric is usually not ...
- 2,129