All Questions
8,725 questions
5
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0
answers
241
views
Roots of unity in algebraic K-theory
For any commutative ring $R$, the tensor product of (finitely generated, projective) $R$-modules equips the algebraic K-theory $K(R) = K_0(R)$ with the structure of a commutative ring with unit.
For $...
- 4,133
3
votes
0
answers
963
views
How to prove that a map is a Serre fibration?
I want to prove that the homotopy groups of some topological space $B$ of interest to me (not a CW complex) are trivial. I have a strategy of proof that consists in introducing another space $E$ that ...
- 14.4k
0
votes
1
answer
200
views
what is the image of $\partial( 1_{S^n})$ for the exact sequence for the fibration $X \to E \to S^n$
what is the image of $\partial 1_{S^n}$ where $\cdots \pi_n(S^n)\rightarrow \pi_{n-1}(X) \rightarrow \pi_{n-1}(B)\rightarrow\cdots$
- 699
3
votes
1
answer
507
views
What are some interesting grading/curving systems you have seen for a course? [closed]
It seems like every math course has something unique in how things are graded.
1) What are some interesting grading systems you have seen/used? (include curving types, etc.)
2) What are some pros ...
-1
votes
1
answer
499
views
How does a chain map induce another chain map on an isomorphic chain complex?
I have 2 ways of defining a chain complex on a manifold, one of which is the cellular chain complex, $C^{CW}_*$. I know that a cellular map $f: X^n \rightarrow Y^n $ such that $ f(X^n) \subset Y^n $ ...
- 1
3
votes
1
answer
928
views
Simple applications of Atiyah-Bott localization
I am looking for some simple and concrete -- but still non-trivial and illustrative -- applications of Atiyah-Bott localization in the context of equivariant cohomology.
Do you know any good ones?
- 21k
7
votes
0
answers
177
views
An explicit description of injective fibrations
If $M$ is a combinatorial model category and $C$ a small category, then the category $M^C$ admits an injective model structure in which the cofibrations and weak equivalences are levelwise. I would ...
- 66.8k
12
votes
0
answers
661
views
Mapping cylinders of fibrations
If $p: E\to B$ is a fibration, is the map $q:M_p \to B$ from the mapping cylinder
of $p$ also a fibration?
I know that it is if $p$ is trivial, or locally trivial; and I know (from Strøm's "The ...
- 12.5k
1
vote
0
answers
222
views
Loops in CW-complexes and the 2-skeleton
Let $X$ be a path-connected CW-complex. If $\omega: [0,1] \to X$ is a loop in $X$ and $\partial$ the boundary operator in simplicial homology, then $\partial(\omega)=\omega(1)-\omega(0)=0$, i.e. $\...
- 16.2k
0
votes
1
answer
225
views
Is the Euler characteristic of a certain nonlinear variety related to that of a certain linear variety?
(This is a generalization of a question I posted a week ago.)
I'm looking at a variety sitting inside the algebraic torus $(\mathbb{C}\setminus 0)^n$ generated by the ideal $I = (*x_1^{\alpha_1} + \...
2
votes
1
answer
329
views
Model categories and cellular maps
A question came up on MSE and it generated, for me, the following question:
When looking at the maps of CW/cell/simplicial complexes do the cellular/simplicial maps have a model theoretic ...
- 3,726
2
votes
1
answer
406
views
Are these systems of linear equations always solvable
Let $X$ be a (finite) simplicial complex and let $f$ be a map from the set of its $n$-Simplices to a abelian group $A$, with the property, that every cycle maps to $0$ (extending $f$ linearly).
Let $...
- 11.1k
2
votes
0
answers
84
views
Zeroth G-equivariant Stable Stem [duplicate]
Let G be a finite group. Can anyone give me a motivation and rigorous proof of the Burnside ring A(G) is isomorphic to the zeroth G-equivariant stable stem ?
- 745
1
vote
1
answer
358
views
What does the weights of Lie group mean?
Let $\Delta=\{\alpha_1,\alpha_2\}$ be the simple root system
of the exceptional Lie group $G_2$
with $\alpha_1$ is short and $\alpha_2$ is long,
so $\lambda_1=2\alpha_1+\alpha_2,\lambda_2=3\...
- 139
9
votes
1
answer
266
views
Branch cuts of $GL_n^+(\mathbb{R})$
Branch cuts
Let $GL_n^+(\mathbb{R})$ denote the group of $n\times n$ real matrices with positive determinant. Topologically, $GL_n^+(\mathbb{R})$ is connected, and
$$ \pi_1(GL_2^+(\mathbb{R})) = \...
- 13k
3
votes
0
answers
431
views
Concrete questions that turn into math problems [closed]
I'm writing an article about the way we teach math, trying to find out why so many people are discouraged from learning, and have no interest for math and logic.
At some point, I want to show that ...
- 131
2
votes
0
answers
285
views
Generators of local homology groups of an isolated critical point
This is a basic Morse theory question:
Let $M$ be a smooth manifold, and $f:M\to\mathbb{R}$ a smooth function with an isolated critical point $x$. Set $c:=f(x)$. The local homology of $f$ is the ...
5
votes
0
answers
571
views
Are there cohomology classes on a hyperkähler manifolds which pull back to the Stiefel-Whitney classes on every Lagrangian submanifold?
This is a bit of a stab in the dark but I was wondering if anyone has defined cohomology classes on a hyperkähler manifold which pull back to the Stiefel-Whitney classes on any submanifold which is ...
- 44.7k
1
vote
0
answers
138
views
G-graphs and Cayley graphs
with which kinds of group we can make a G-graph(Bretto 2011) which are hamiltonian Cayley graph?
2
votes
0
answers
131
views
Topological dimension of quotient group determined by the inverse limit of discrete free monoids
Must the natural quotient group of the inverse limit of a sequence of nested discrete free monoids have topological dimension zero?
The question might well be open, but I would be grateful for news ...
- 1,968
8
votes
0
answers
373
views
Completion of n-fold Segal spaces
During a reading course about Jacob Lurie's paper about Local TQFTs, I came across the notion of Segal spaces as models for $(\infty,1)$-categories. Looking at things like the "double nerve" for 2-...
- 7,637
0
votes
0
answers
308
views
A modified version of K-theory for manifolds ?
If $X$ is a compact smooth manifold, $K^{0}(X)$ can be defined as the algebraic $K_{0}$-group of $C^{\infty}(X)$. In order to do that we use the following equivalence relation: we say that two ...
- 577
1
vote
1
answer
127
views
Bibliographical reference needed (characterizing the weak equivalences of a model category)
I need a bibliographical reference for this fact: let $\mathcal{M}$ be a model category such that all objects are cofibrant; then the class of weak equivalences is the class of maps f such that $\...
- 3,901
1
vote
0
answers
241
views
Two-point desuspension for augmented chain complexes?
Let $X$ be a chain complex augmented over $\mathbf{Z}$ with augmentation $\varepsilon_X:X_0 \to \mathbf{Z}$. We define $[1](X)$ to be the $\mathbf{Z}$-augmented chain complex such that $[1](X)_0 = \...
- 19.6k
0
votes
0
answers
381
views
Is the cap product bilinear?
This is probably a stupid question, so I apologize in advance.
On p. 239 of Hatcher's book, he defines the cap product $C_k(X;R)\times C^l(X;R)\to C_{k-l}(X;R)$ for $k\geq l$, which he claims is $R$-...
- 1
9
votes
0
answers
606
views
Historical and terminological questions about Dan Kan's Ex functor and its relation to the classical case of simplicial complexes
Recall that we may define a functor $\xi:\Delta\to \operatorname{Poset}$ sending a simplex $[n]$ to the set of monotone injections $[k]\hookrightarrow [n]$ for $k\geq 0$ (effectively, $k\leq n$ as ...
5
votes
0
answers
583
views
Cohomology of Real algebraic Varieities
I understand Serre's GAGA theorem as saying that equations over algebraically closed fields can be studied equally from the algebraic and analytic points of view, at least with respect to cohomology.
...
user2529
5
votes
1
answer
466
views
nerves of crossed complexes, group T-complexes and classifying spaces
A (reduced) crossed complex is, intuitively, a non-abelian complex of groups $\ldots \to G_2 \to G_1 \to G_0$ with a $G_0$ action in such that $G_1 \to G_0$ is a crossed module.
There are a couple of ...
- 35.5k
2
votes
1
answer
412
views
How can I prove that the derived couple of the homotopy exact couple is an invariant?
I'm working on (yet) an(other) exercise from Mosher & Tangora's "Cohomology Operations and Applications to Homotopy Theory". This one is about the homotopy exact couple, which is defined for a ...
- 6,069
3
votes
1
answer
210
views
Classifying space of variant on category of simplices
This question might have an easy answer, but my research is far from the region of topology that makes use of classifying spaces of categories, so I can't find it.
For (possibly infinite) integers $0 ...
2
votes
0
answers
179
views
About Thom Theorem (representation submanifold for $H_{n-2}(M^n)$)
Recall Thom theorem : If $M^n$ is a smooth orientable closed manifold then any homology class in $H_{n-2}(M)$ is represented by the fundamental class of a smooth submanifold.
And in the Harper and ...
- 1,100
19
votes
0
answers
773
views
Folk Functorial Figuring
In the CRM Proceedings & Lecture Notes Volume 50 "A Celebration of the Mathematical Legacy of Raoul Bott" Herbert Shulman writes (p. 48):
"[Bott] taught many of us to think functorially, like ...
- 2,684
4
votes
1
answer
497
views
Riemann Existence Theorem for Real Curve
By real curve, we mean a Riemann surface $X$ together with an anti-holomorphic involution
$\sigma : X\rightarrow X$. Let $S$ be a finite subset of $X$. For each point $x\in S$, we associate a positive ...
- 135
2
votes
1
answer
218
views
Shrinkable maps and universal weak equivalences
Recall that a morphism $f:X \to B$ is called shrinkable is there exists a section $s:B \to X$ together with a homotopy $$H:I \times X \to X$$ from $sf$ to $id_X$ over $B,$ i.e. for all $t$, the map $$...
- 15.5k
2
votes
0
answers
176
views
Two questions on axiomatic homology
1) Given the Eilenberg-Steenrod axioms, there are several Mayer-Vietoris type sequences that can be deduced. The most general form seems to be
$$\rightarrow H_n(X \cap Y, A \cap B) \rightarrow H_n(X,...
- 245
10
votes
0
answers
235
views
Computation of stable cohomology ring of SL_n(Z) using algebraic topology
It is known that $H^k(SL(n,\mathbb{Z}))$ is independent of $n$ for $n \gg k$, so we can define a stable cohomology ring
$$V = \text{lim}_{n \rightarrow \infty} H^{\ast}(SL(n,\mathbb{Z});\mathbb{R}).$$...
- 101
4
votes
1
answer
360
views
An analogue of Lefschetz hyperplane theorem for complements to subvarieties in $\mathbb C^n$ ?
Let $V^{2k}$ be a complex subvariety of dimension $2k$ (real dimension $4k$) in $\mathbb C^n$. Let $A$ be a complex $n-k$ dimensional plane in $\mathbb C^n$.
Question. Is it true that the inclusion $...
- 14.3k
4
votes
1
answer
314
views
Homology of a complex projective conic
Let $Q$ be a smooth conic (the zero set of a homogeneous degree 2 polynomial)
in $\mathbb{P}^2(\mathbb{C})$ and let $j:Q\rightarrow\mathbb{P}^2(\mathbb{C})$ be the
immersion in the projective plane. ...
- 1,727
0
votes
0
answers
292
views
Is $GL_{S^1}(H)$ (for $H$ Hilbert space ) path-connected?
Due to the negative answer to my last question I want to know if at least the following is true:
Let H be an infinite dimensional separable complex Hilbert space with $S^1$-action. Let $\text{Gl}_{S^...
- 1,282
6
votes
0
answers
398
views
Differential forms on the simplex which are "constant towards the boundary"
Let $\Delta^k$ the standard $k$-dimensional simplex, $\Delta^k=\{(x^0,\dots,x^k)\in \mathbb{R}^{k+1}|\sum_{i=0}^kx^i=1\}$, and let $\Omega^\bullet(\Delta^k)$ be the de Rham complex of smooth ...
- 6,777
4
votes
1
answer
195
views
Contractible space of maps between Eilenberg-Mac Lane spaces, 2
Let $G$ and $H$ be torsion abelian groups.
Are the following are equivalent:
$\mathrm{Hom}(G, H) = 0$
$\mathrm{map}_*( K(G, m), K(H,n)) \sim *$ for all $n, m\geq 1$
?
Clearly (2) implies (1).
- 12.5k
5
votes
1
answer
383
views
Killing Chern classes
Let $G$ be a compact connected Lie group and let $E\to B$ be a principal $G$-bundle. Suppose $a$ is a rational cohomology class of $E$ such that its pullback $b$ under an orbit inclusion map $G\to E$ ...
- 23.5k
4
votes
1
answer
328
views
Algebraic K-groups and braids
This is (I think) a reference request:
Are there calculations of any algebraic K-groups for the (group ring of) the Artin braid groups?
- 1,180
5
votes
0
answers
263
views
Coloring $\mathbb{Z}^k$ and a fixed point theorem
This is potentially another approach to this question. I put it as an update there, but perhaps it would be better to post it separately. If we color $\mathbb{Z}^k$ with the $\ell_\infty$ metric in ...
user6976
2
votes
1
answer
270
views
homotopy groups for good rings
I think this question should already be abound in literature but the only place I find is from this article:
http://math.uchicago.edu/~lxiao/files/notes/Fundamental%20Groups.pdf
which seems to be ...
- 799
2
votes
1
answer
307
views
Name for a kind of topological property?
What should I call a property (P) of (open) subspaces of a space $X$ such that:
If $U$ satisfies (P), then so does every open subset $V\subset U$
If {$U_i$} is a pairwise disjoint collection of ...
- 12.5k
4
votes
0
answers
732
views
Spectral sequence for reduced homology
In the Serre spectral sequence, is it true that we can replace homology by reduced homology? That is:
If $f:X\rightarrow B$ is a Serre fibration,with $F$ the fiber, then if
$\tilde E^2_{pq}=\tilde H_p(...
- 1,499
0
votes
0
answers
127
views
How to compute the Betti numbers of S-D for a surface S and a divisor D?
Let S be a projective non-singular surface and D a Cartier divisor which has a smooth representative. Can the Betti numbers of S-D be represented by the Betti numbers of S and D? In a paper $b_i(S-D)=...
- 1
10
votes
0
answers
464
views
Bicommutative Hopf algebras have internal hom objects. What are they?
Let $k$ be a field and $\mathcal H$ the category of conilpotent bicommutative Hopf algebras over $k$. (Conilpotent means that every element eventually becomes zero if you apply the reduced ...
- 6,162
2
votes
1
answer
215
views
$b_2$ of the blow up of a complex $3$-fold in a curve
Suppose that $V$ is a complex analytic manifold of dimension 3 with mild singularities, say it is an orbifold (i.e. has only quotient singularities). Let $C$ be a complex irreducible curve in $V$. ...
- 14.3k