Questions tagged [at.algebraic-topology]
Homotopy theory, homological algebra, algebraic treatments of manifolds.
2,307
questions with no upvoted or accepted answers
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When is a compact topological 4-manifold a CW complex?
Freedman's $E_8$-manifold is nontriangulable, as proved on page (xvi) of the Akbulut-McCarthy 1990 Princeton Mathematical Notes "Casson's invariant for oriented homology 3-spheres".
Kirby showed that ...
- 3,851
29
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0
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935
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Todd class as an Euler class
Let $X$ be a relatively nice scheme or topological space.
In various physics papers I've come accross, the Todd class $\text{Td}(T_X)$ is viewed as the Euler class of the normal bundle to $X\to LX$. ...
- 5,565
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Why do polytopes pop up in Lagrange inversion?
I'd be interested in hearing people's viewpoints on this. Looking for an intuitive perspective. See Wikipedia for descriptions of polytopes and the Lagrange inversion theorem/formula (LIF) for ...
28
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On the (derived) dual to the James construction.
Background
If $X$ is a based space then the James construction on $X$ is the space $J(X)$ given by
$$
X \quad \cup \quad X^{\times 2} \quad \cup \quad X^{\times 3} \quad \cup \quad \cdots
$$
in ...
27
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0
answers
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Spectral sequences as deformation theory
I believe that running the spectral sequence of a filtered complex / spectrum $ \cdots \to F_n \to F_{n+1} \to \cdots$ can be viewed as doing deformation theory in some very primitive "derived ...
- 62.6k
27
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Computational complexity of topological K-theory
I am a novice with K-theory trying to understand what is and what is not possible.
Given a finite simplicial complex $X$, there of course elementary ways to quickly compute the cohomology of $X$ with ...
- 666
25
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0
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Does the Tate construction (defined with direct sums) have a derived interpretation?
Any abelian group M with an action of a finite group $G$ has a Tate cohomology object $\hat H(G;M)$ in the derived category of chain complexes. There are several ways to define this. One is as the ...
- 51.9k
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The $(\infty, 1)$-category of all topological spaces, including the bad ones
[Edit: Corrected some false claims and modified questions accordingly.]
Let $\mathcal{S}$ be the cocomplete $(\infty, 1)$-category generated by a point.
This is conventionally known as the $(\infty, 1)...
- 15k
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p-Adic String Theory and the String-orientation of Topological Modular Forms (tmf)
I am going to ask a question, at the end below, on whether anyone has tried to make more explicit what should be, it seems to me, a close relation between p-adic string theory and the refinement of ...
- 19.7k
23
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460
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Topological loops vs. algebro-geometric suspension in Hochschild homology
Let $k$ be a base commutative ring, and let $A$ be a (unital but not necessarily commutative) $k$-algebra. The cone on $A$ is the ring $CA$ of infinite matrices $(a_{ij})_{i,j \geq 1}$ that are ...
- 6,049
23
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577
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What is the symmetric monoidal functor from Clifford algebras to invertible K-module spectra?
There ought to be a symmetric monoidal functor from the symmetric monoidal $2$-groupoid whose objects are Morita-invertible real superalgebras (precisely the Clifford algebras), morphisms are ...
- 116k
23
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0
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783
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Characteristic classes for $E_8$ bundles
$\DeclareMathOperator\B{B}\DeclareMathOperator\SU{SU}$Given a principal $E_8$ bundle $P\rightarrow X$ one can take the
adjoint representation $\rho :E_8\rightarrow \SU(\mathbb C^{248})$
and form the ...
- 694
22
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0
answers
410
views
Unoriented bordism and homology, reference?
The following has undoubtedly been known to the experts for years, but I only noticed it the other day. Can anyone give a reference?
One can prove Thom's theorem to the effect that every mod $2$ ...
- 55.1k
22
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0
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853
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Bar construction vs. twisted tensor product
One may study the cohomology of a space $E$ expressed as a homotopy pullback of $X$ and $Y$ over $Z$ using either the Eilenberg-Moore spectral sequence or the Serre spectral sequence for the fibration ...
- 985
22
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666
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Are there "chain complexes" and "homology groups" taking values in pairs of topological spaces?
Throughout this question, notation of the form $(X,A)$ denotes a sufficiently nice pair of topological spaces. I think for most of what I'm saying here, it is enough to assume that the inclusion $A \...
- 15.5k