Questions tagged [at.algebraic-topology]
Homotopy theory, homological algebra, algebraic treatments of manifolds.
8,351
questions
4
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1
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Detailed exposition of construction of Steenrod squares from Haynes Miller's book
$\DeclareMathOperator\Sq{Sq}$Chapter 75 of Haynes Miller's book on algebraic topology contains a beautiful construction of the Steenrod squares $\Sq^i$.
Roughly speaking, it goes as follows. All ...
- 41
4
votes
1
answer
148
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When can a generalized connected sum be aspherical
Let $M$ and $N$ be compact $n$-dimensional manifolds with a common (nicely embedded) compact submanifold $S$ (we may assume that the inclusion of $S$ in $M$ and $N$ is $\pi_1$-injective). Let $M\#_S N$...
- 151
3
votes
0
answers
49
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Rational model for composition of linear isometries
There is a composition map on spaces of linear isometries (over $\mathbb{C}$ say)
$$
\mathcal{L}(\mathbb{C}^k, \mathbb{C}^\ell) \times \mathcal{L}(\mathbb{C}^\ell, \mathbb{C}^m) \longrightarrow \...
- 781
25
votes
8
answers
3k
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Object of proven finiteness, yet with no algorithm discovered?
I explain my title by two examples in number theory:
The rational points on elliptic curve over number fields forms a finitely generated abelian group, so its rank is an integer, but so far we do not ...
- 1,033
6
votes
0
answers
102
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Eulerian posets and order complexes
To every poset $P$ it is possible to associate its order complex $\Delta(P)$. The faces of $\Delta(P)$ correspond to chains of elements in $P$. An Eulerian poset is a graded poset such that all of its ...
- 1,949
-3
votes
0
answers
54
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How to show H_BM(Conf_k(D_n))=H_BM(Conf_k(I_n))
How to show H_BM(Conf_k(D_n))=H_BM(Conf_k(I_n))?
I am trying to prove that the Borel-Moore homology of n-punctured disk D_n is the same as the Borel-Moore homology of interval [0,1] but with n-points ...
6
votes
0
answers
171
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Twisting cochain intuition
I'm currently reading through Ed Brown's paper "Twisted tensor products, I", (MR105687, Zbl 0199.58201) and I couldn't find any simple examples of twisting cochains. I understand all ...
- 161
1
vote
0
answers
65
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End cohomology and space of ends
I have started learning about end cohomology, and as far as I understand, the zeroth end cohomology $H_e^0(M; \mathbb{Z})$ is isomorphic to the zeroth Čech cohomology $\check{H}^0(e(M); \mathbb{Z})$, ...
- 1,097
4
votes
0
answers
93
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$E_k$-operads and actions on objects inside $k$-tuply monoidal $n$-category
I understood more or less the claim that $k$-tuply monoidal $n$-categories can be seen as $n$-categories equipped with an action of the $E_k$-operad.
For $k=2$, we have a homotopy equivalence $E_2(r) \...
- 1,743
2
votes
0
answers
70
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Action of $V$ on the homology of a subposet of the poset of affine subspaces of $V$
Let $(V,Q)$ be a pair, with $V=\mathbb{F}_2^{2n}$ ($n\geq 2$) and $Q$ a nondegenerate quadratic form on $V.$ We consider the poset $\mathcal{P}_n$ of affine totally isotropic (with respect to $Q$) ...
- 215
4
votes
1
answer
214
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Intersection pairing on non-compact surface
Let $S$ be a smooth oriented connected $2$-manifold. We have an algebraic intersection pairing $\omega\colon H_1(S) \times H_1(S) \rightarrow \mathbb{Z}$. If $S$ is compact, then this is ...
- 43
1
vote
0
answers
55
views
Which sheaves are good for calculating extraordinary restriction?
Let $X$ be a sufficiently nice locally compact Hausdorff space and let $i:Y\subset X$ be the inclusion map of a sufficiently nice closed subspace. For example, one could take $X$ to be a locally ...
- 23.4k
5
votes
0
answers
143
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$\infty$-category of spectra and cofibrancy
I have two options for the $\infty$-category of spectra. I would like to know they are equivalent as $\infty$-categories.
Premise: by work of Dwyer and Kan, if we have a simplicial model category, the ...
- 412
4
votes
1
answer
125
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Are monomorphisms in an $\infty$-topos preserved by $0$-truncation?
Let $\mathfrak{X}$ be an $\infty$-topos and let $f\colon X\to Y$ be a morphism of $\mathfrak{X}$. We say that $f$ is a monomorphism if it is $(-1)$-truncated which means that for every $Z\in\mathfrak{...
- 10.3k
8
votes
1
answer
216
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Product structure in Milnor exact sequence
Let $h^*$ be a (multiplicative) generalized cohomology theory. Let $X$ be a CW complex which is a union of an increasing sequence $X_0 \subset X_1 \subset X_2 \subset \cdots$ of subcomplexes. Then ...
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