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Results tagged with homotopy-theory
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user 41075
Homotopy theory is an important sub-field of algebraic topology. It is mainly concerned with the properties and structures of spaces which are invariant under homotopy. Chief among these are the homotopy groups of spaces, specifically those of spheres. Homotopy theory includes a broad set of ideas and techniques, such as cohomology theories, spectra and stable homotopy theory, model categories, spectral sequences, and classifying spaces.
1
vote
1
answer
303
views
isotopy equivalence (topological meaning) between $CW$-complexes
Let $M$ and $N$ be $CW$-complexes.
Definition. (different from the isotopy notion in geometry of submanifolds). A (topological) isotopy is a fibre-wise continuous map
$$
F: M\times [0,1]\longright …
- 1,990
5
votes
1
answer
404
views
triviality of homology with local coefficients
Let $X$ be a manifold or a CW-complex.
Let
$\pi: \tilde X\longrightarrow X$
be a covering map.
Let $\pi_1(X)$ be the fundamental group of $X$ and let $\rho: \pi_1(X)\longrightarrow O(n)$ be an ort …
- 1,990
0
votes
1
answer
188
views
cohomology algebra of unordered configuration space on Euclidean space
In Homology of $C_{n+1}$-spaces, $n\geq 0$, F.R. Cohen, Lecture Notes in Mathematics, Vol. 533, page 210 (the preface part before contents):
Line 2: ... is used to compute the precise algebra struct …
- 1,990
2
votes
1
answer
336
views
fiber sequence of principal bundles
Let $G$ be a group, either a Lie group or a discrete group. Let a principal $G$-bundle
$$
G\to E\to B,$$
then $B=E/G$, the orbit space under action of $G$.
Let $BG$ be the classifying space of $G$. …
- 1,990
1
vote
1
answer
292
views
Unordered configuration space of $\mathbb{R}P^1$
In the paper
GEOMETRY OF TRUNCATED SYMMETRIC PRODUCTS AND REAL
ROOTS OF REAL POLYNOMIALS, JACOB MOSTOVOY, Bull. London Math. Soc. (1998) 30 (2):
159-165,
Theorem 2. (b): $TP^n(\mathbb{R}P^1)$ is ho …
- 1,990
1
vote
1
answer
223
views
can we take skeletons of covering maps to give new covering maps?
Let $X$ be an $n$-dimensional cell complex.
We attach an $(n+1)$-cell $e^{n+1}$ to $X$ and obtain a new cell complex $X'$.
Take the universal cover (or a general covering space) $\tilde X'$ of $X'$.
W …
- 1,990
9
votes
1
answer
540
views
cohomology of classifying space of permutation groups
Let $\Sigma_k$ be the permutation group of order $k$.
Let $r: \Sigma_k\to GL(k)$ be the regular representation by permuting the order of the standard basis of $\mathbb{R}^n$.
Let $\rho: B\Sigma_k\t …
- 1,990
4
votes
1
answer
600
views
Relation between cohomology of ordered and unordered configuration spaces
Let $M$ be a manifold. Then $F(M,k)/\Sigma_k$, the unordered configuration space of $k$ points, is obtained as a quotient of $F(M,k)$, the ordered configuration space of $k$ points, by the group actio …
- 1,990
4
votes
1
answer
419
views
homology of configuration spaces of non-compact manifolds
Let $M$ be a manifold.
Let $F(M,n)$ be the configuration space of $n$-tuples on $M$.
Let $B(M,n)=F(M,n)/S_n$, where $S_n$ is the symmetric group of order $n$, be the corresponding unordered config …
- 1,990
7
votes
2
answers
720
views
contractible configuration spaces
Let $F(M,k)=\{(x_1,\cdots,x_k)\mid x_1\cdots,x_k\in M,x_i\neq x_j, \text{ for } i\neq j \}$. It is known that $F(\mathbb{R}^\infty,k)$ is contractible for each $k$.
My question: is $F(S^\infty,k)$ …
- 1,990
1
vote
1
answer
257
views
permutation action on cohomology of Stiefel manifolds
Let $V_k(\mathbb{R}^n)$ be Stiefel manifolds.
In the paper The cohomology rings of real Stiefel manifolds with integer coefficients, Martin Čadek, Mamoru Mimura, and Jiří Vanžura, J. Math. Kyoto …
- 1,990
3
votes
1
answer
364
views
cohomology ring of configuration spaces
In the paper configuration spaces: applications to Gelfand-Fuks cohomology, by F. Cohen and L. Taylor, Bull. Amer. Math. Soc., 1978, theorem 1, I did not find the proof. What method did the author use …
- 1,990
0
votes
1
answer
775
views
unordered configuration space of pointed space
Let $(X,*)$ be a pointed topological space.
Let $F(X,k)=\{(x_1,\cdots,x_k)\in X^k\mid \forall i\neq j: x_i\neq x_j, \}$.
Let $F(X,k)/S_k$ be the $k$-th unordered configuration space.
Is there an i …
- 1,990
-2
votes
1
answer
290
views
stable splitting into a wedge sum [closed]
Suppose $X$ is a CW-complex such that there is a stable splitting of $X$ into wedge sum
$$
\Sigma^t X\cong \bigvee _{k=1}^\infty Y_k.
$$
(1). Does this imply
$$
X\to \Sigma^tX\to \bigvee _{k=1}^\inf …
- 1,990
0
votes
1
answer
174
views
iterated loop spaces and configuration spaces [closed]
In the lecture notes by J.P. May, The geometry of iterated loop spaces, Chapter 5, formula (1), (2) and (10), a map
$$
\phi: Hom_T(X,\Omega Y)\to Hom_T(SX,Y)
$$
is defined. And a map
$$
\eta_n=\phi^{ …
- 1,990