All Questions
9,056 questions
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group actions on fibre bundles
Suppose that we have a group $G$ acting on the spaces $E$ and $B$. Suppose moreover that we have fibre bundles $\xi$ and $\eta$ in the following commutative diagram
If $\xi$ is a trivial bundle, i.e....
- 1,990
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votes
1
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417
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fiber, homotopy fiber of spaces
Suppose we have a pullback of topological spaces (CW-complexes) $B\rightarrow A\leftarrow C$ which I will denote by $D$.
Assumptions
The induced map $D\rightarrow C$ is a trivial fibration
The map $...
- 1,974
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votes
1
answer
779
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 ...
- 1,990
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1
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749
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Pushout of categories along embeddings gives homotopy pushout?
Consider the pushout of a diagram $A\leftarrow B\rightarrow C$ of categories and assume that at least one of the arrows is an embedding, i.e. injective on objects and arrows. When applying the nerve ...
- 1,550
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1
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642
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fibre bundle as a boundary of a fibre bundle
Let $M_{n+1}$ be a fibre bundle with $S_1$ as the base and $n$-dimensional CW complex $F_n$ as the fibre.
Assume $M_{n+1}$ is oriented.
(1) Can one show that $M_{n+1}$ is always a boundary of a CW ...
- 4,826
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1
answer
403
views
When does a power semigroup have a zero, and what can the zero be?
Let $S$ be a semigroup. The power semigroup of $S$ is the set $P(S)=2^S\setminus\lbrace\varnothing\rbrace $ with the operation $$AB=\lbrace ab\ |\ a\in A,\ b\in B\rbrace.$$
This operation is ...
- 1,445
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1k
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about relative homotopy group
$S(RP^2),S(CP^2)$denote suspension of real and complex projective space.
Then are the first order relative homotopy group $\pi_1(S(RP^2),RP^2),\pi_1(S(CP^2),CP^2) $trivial?Why?
- 689
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1
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600
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Continuous variation from solution of easy problem to solution of hard problem
I asked this question a week ago over on math.stackexchange and got no reply, so I am asking here with slightly different wording. I am trying to prove that there exists a solution to a problem. I ...
- 298
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1
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73
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If $\widehat{\Gamma}$ is a simply connected clique complex then $\mathrm{Out}(A_\Gamma)$ is an infinite group
Let $\Gamma$ be a simplicial graph and $\widehat{\Gamma}$ be the corresponding clique complex (the flag complex obtained after adding simplices for each compete graph). We can costruct the right-...
- 911
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1
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272
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When do cobordism groups depend on differential structure? [closed]
I heard that cobordism group with structures sometimes depend on differential structure of space.
Do you know any examples or references about this facts?
I want to know when difference occur between ...
- 1
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1
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177
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Existence of certain continuous curves
Let $\mathbb{S}^n$ be the $n$-sphere. I would like to know if anyone knows of the following result in the literature (or whether anyone knows a proof/counterexample).
Let $f\colon\mathbb{S}^1\times[0,...
- 1,453
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2
answers
284
views
Motivation and reference for Brauer algebras
I am looking for a good reference and motivation for Brauer monoid and Brauer algebras. Kindly help me with some suggestions. Thanks.
- 141
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877
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Are all manifolds in $\mathbb R^n$ homeomorphic to a smooth manifold?
My question is whether all manifolds that can be embedded in $\mathbb R^n$ are homeomorphic to a smooth manifold?
I know that every smooth manifold can be triangulated which I think is a result of ...
- 4,862
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1
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170
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The kernel of the natural map $\pi_k(BU(r)) \to \pi_k (BU)$
Is this group known outside of the stable range? If so, what is it? If not, what is known about it?
- 13
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486
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History of Poincare conjecture in higher dimension [closed]
As far as I know, when Poincare formulated his well known conjecture, the original statement was the follwoing: if a closed manifold has the same homology groups as the sphere it is homoeomorphic to ...
- 9,330
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511
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Principal bundle associated to a fiber bundle
Let $\pi : E\to B$ be a fiber bundle with fiber $F$ over a finite complex $B$ whose structure group is a compact Lie group $G$. How can we determine the principal $G$-bundle associated to $\pi$? For ...
- 45
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368
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Formal Space and Rational Homotopy Theory [closed]
Please give me the proof that for a formal space Massey triple products vanish.
- 141
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1
answer
483
views
Is this manifold orientable? [closed]
Let $C$ be the set of points $(a,b,c,d) \in \mathbb{C}^4$ which satisfy
1) $ \left|a\right|^2+\left|c\right|^2=\left|b\right|^2+\left|d\right|^2 =1 $.
2) $ a\bar{b}+c\bar{d}=0 $
There is a (...
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1
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170
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Is there a Lie group which is made $S^n \cup_f ~e^m$?
Is there a Lie group G which has only two cells?
i.e. $ G = S^n \cup_f~ e^m$ where $f:S^{m-1}\to S^n$ with $m>n$
How many exists that groups? Infinitely many?
If there is no Lie group which has ...
- 699
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1
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225
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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} + \...
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1
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423
views
What Is This Quotient Space?
Let $X$ be a finite CW-complex with only even cells $x_1,\ldots, x_k$ and let $Y$ be the complex obtained by attaching one more even cell to $X$, call it $y$. Assume both $X$ and $Y$ are connected. ...
- 61
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65
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Is there a characterization of monoids that distribute over each other?
Let $(M, e_1, \times_1, e_2, \times_2)$ be an algebraic structure such that
$(M, e_1, \times_1)$ and $(M, e_2, \times_2)$ are monoids
$x \times_1 (y \times_2 z) = (x \times_1 y) \times_2 (x \times_1 ...
- 631
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328
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Relationship between quotient CW-complexes after attaching cells
I have been trying to prove the following simple-looking result which I require for some work in low-dimensional topology. I expect it is likely true and in a textbook somewhere so any reference or ...
- 187
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170
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Isn't every algebraic operad equipped with a trivial weight?
In Loday–Vallette "Algebraic Operads" they state the following result (Theorem 6.6.2, Operadic twisting morphism fundamental theorem):
Let $P$ be a connected weight graded differential ...
- 215
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1
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376
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Relation between trivial tangent bundle $\Leftrightarrow$ certain characteristic classes of tangent bundle vanish [closed]
We know that
framing structure means the trivialization of tangent bundle of manifold $M$.
string structure means the trivialization of Stiefel-Whitney class $w_1$, $w_2$ and half of the first ...
- 447
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1
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502
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Is every compact contractible subset of $\mathbb{R}^n$ homeomorphic to a closed ball of some dimension? [closed]
Question: Is every compact contractible subset of $\mathbb{R}^n$ homeomorphic to a closed ball of some dimension?
This post doesn't quite answer my question because it is about open sets.
- 637
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2
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11k
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Mathematics Roadmap [closed]
I immediately apologize for my English, Google translator is my assistant. I couldn't find the information in my own language.
My question is addressed to people who understand mathematics. I hope for ...
- 25
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408
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How do I show that any finite-dimensional (absolute) CW-complex $X$ is locally contractible?
I know that it holds even if $X$ has infinite dimension, but I am looking for a specific argument in the finite-dimensional case.
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676
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Second homotopy group of the wedge sum of $S^2$ with the presentation complex of a finitely generated group
I am reading a paper which makes the following claim:
let $G$ be a finitely presented group, and let $X$ be the presentation complex of $G$.
Let $X' = X \vee S^2$ be the wedge sum of $X$ with the ...
- 353
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1
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776
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Homology of Hirzebruch surfaces
Let $\mathbb{F}_n:=\mathbb{P}(\mathcal{O}(-n)\oplus\mathcal{O}(0))$ be the $n$th Hirzebruch surface, where $\mathcal{O}(k)$ is the canonical line bundle on $\mathbb{P}^1_\mathbb C$, for any $k\in\...
- 1,252
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143
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Trivialize a cup-product 2-cocycle of $G$ in a larger group $J$
I like to ask a simple question: how to trivialize a cup-product 2-cocycle of $G$ into a 2-coboundary of $J$ in a larger group $J$.
Let us take a nontrivial 2-cocycle $\omega_3^G(g_a, g_b) \in H^2(G,\...
- 755
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2
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662
views
Relation between different definitions of homotopy
When I did a course in topology, we defined "homotopy" in the following way:
"A homotopy between two continuous functions $f$ and $g$ from a topological space $X$ to a topological space $Y$ is ...
- 9
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2
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1k
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When is the Thom class the Poincare dual of the zero section?
As the title suggests, when is the Thom class the Poincare dual of the zero section? For starters, it's true for the normal bundle of an immersion...
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1
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390
views
Moment maps and flat degenerations of toric varieties
We have a flat family of projective varieties with a torus $T$ action, over $\mathbb{A}^1$.
How do the moment map images of the fibers change when we pass from the generic fiber to the special fiber ...
- 1,719
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2
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293
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Restriction of a line bundle to a two-cycle
I am reading a paper on Chiral Differential Operators
http://arxiv.org/pdf/hep-th/0604179v3.pdf
and it says on page 23 that a line bundle over a manifold C can be characterized completely by its ...
- 17
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1
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128
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What are the semigroups in which congruence classes can be multplied like sets?
For a semigroup $S$ and a congruence $\rho$ on $S$, let's say that $\rho$ is good when for all $a,b\in S$ we have that $[ab]=[a][b],$ where $[x]$ denotes the congruence class of $x$ modulo $\rho$ and ...
- 1,445
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1
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385
views
No fixed components in the linear system of the line bundle generating $Pic(X)$
Let $X$ be a K3 surface. I know that $Pic(X)\simeq H^{1,1}(X,\mathbb{Z}):=H^{1,1}(X)\cap H^2(X,\mathbb{Z})$. Let's suppose that $H^{1,1}(X,\mathbb{Z})$ is one dimentional and generated by $\omega=c_1(...
- 107
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673
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(Homotopy) Y ENR and contractible subset implies Y is a retract
I'm trying to solve the following question:
Suppose $Y \subset R^n$ is a Euclidean neighborhood retract. I want to prove that if $Y$ is contractible, then it is a retract of $R^n$.
- 11
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270
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Recovering torsion in singular homology from cplx of singular chains
For a simply connected simplicial complex, a theorem of Whitehead (Derived categories for the working mathematician, bottom of page 2) explains that the associated chain complexes with coefficients in ...
- 3,555
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1
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246
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Noncontractible domain with trivial cohomology [closed]
Can you exhibit an example of a noncontractible domain in R^n with the d-th cohomology groups trivial for all d greater or equal to 1?
Thank you
- 129
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1
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156
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Calculation of L2-dimension
For a group $G$, can we calculate $dim^{(2)}_{\mathcal{N}G}(\ell^2 G)$, where $\mathcal{N}G$ is the von Neumann algebra of $G$ and $\ell^2 G$ is the Hilbert space on $G$? I want to see whether this is ...
- 537
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1
answer
317
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Embedding of $T^{2}$ on $S^{1}\times S^{2}$.
Let $i:T^{2}\rightarrow S^{1}\times S^{2}$ be a embedding map and $i_{*}:\pi(T^{2})\rightarrow \pi(S^{1}\times S^{2})$ be the homomorphism induced by $i$. If $i(T^{2})$ is separable in $S^{1}\times S^{...
- 45
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1
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264
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When does nerving the groupoid completion of a category give a weak equivalence?
In an answer to another of my questions, Spice the Bird explains that for any monoid $M$, the map $NM\to NK(M)$ is a weak equivalence. Here $N:{\mathsf{Cat}}\to {\mathsf{sSet}}$ is the nerve functor ...
- 681
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1
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252
views
the homotopy type of a product of some spaces
let S be the n-sphere.
how can we see that
(SxS) smash S
has the homotopy type of a wedge of spheres?
- 11
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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
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2
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641
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Looking for general approaches to show connectedness of topological groups
Let $G$ be a topological group. One general approach to show that $G$ is connected is the following:
For every subgroup $H\leq G$ (not necessarily closed) we have a projection map:
$$
\pi: G\...
- 7,609
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1
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427
views
what does the coefficients ring of generalized cohomology defined by the unitary Thom spectrum like?
Let $MU$ be the unitary Thom spectrum, then it gives a generalized cohomology,
so what is the coefficients $MU^*(point)$ like?
Is it just the complex cobordism ring $\Omega_U^*?$
- 139
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votes
1
answer
91
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Topological Properties of Subsets of $R^{m}$ induced by Smooth Manifolds in Matrix Spaces
We know that $M_{m \times n } $ is isomorphic to $R^{mn}$. Let's take a smooth manifold $\mathbb{M}$ in $R^{mn}$ and fix a point in $R^{n}$, say, $p$. Now realize the manifold $\mathbb{M}$ as a subset ...
- 101
0
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1
answer
231
views
Explaining some detail in Wall's paper of CW-complexes
For a given map $\phi :X\longrightarrow Y$, the mapping cylinder of $\phi$ is defined by $M_{\phi}:=Y\cup_{\phi} (X \times \{ 1\})$. Denote $\pi_n (M_{\phi},X \times \{ 1\} )$ by $\pi_n (\phi)$...
- 1,182
0
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1
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239
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Understanding the definition of left homotopy as given in Quillen’s Homotopical algebra book
Given two topological spaces $X,Y$, and two maps $f,g:X\rightarrow Y$, there is a notion of homotopy between $f$ and $g$. It is given by a continuous map $H:X\times I\rightarrow Y$ such that the ...
- 6,023