Let $X$ be a CW complex and $Y$ a CW subcomplex. If $X$ has no cell of dimension $n$, for some $n>0$, then $X/Y$ has no cell of dimension $n$. Is it true? Why?
2026-03-27 03:49:13.1774583353
cells of quotient CW complex
913 Views Asked by Bumbble Comm https://math.techqa.club/user/bumbble-comm/detail At
1
There are 1 best solutions below
Related Questions in GENERAL-TOPOLOGY
- Is every non-locally compact metric space totally disconnected?
- Let X be a topological space and let A be a subset of X
- Continuity, preimage of an open set of $\mathbb R^2$
- Question on minimizing the infimum distance of a point from a non compact set
- Is hedgehog of countable spininess separable space?
- Nonclosed set in $ \mathbb{R}^2 $
- I cannot understand that $\mathfrak{O} := \{\{\}, \{1\}, \{1, 2\}, \{3\}, \{1, 3\}, \{1, 2, 3\}\}$ is a topology on the set $\{1, 2, 3\}$.
- If for every continuous function $\phi$, the function $\phi \circ f$ is continuous, then $f$ is continuous.
- Defining a homotopy on an annulus
- Triangle inequality for metric space where the metric is angles between vectors
Related Questions in GEOMETRY
- Point in, on or out of a circle
- Find all the triangles $ABC$ for which the perpendicular line to AB halves a line segment
- How to see line bundle on $\mathbb P^1$ intuitively?
- An underdetermined system derived for rotated coordinate system
- Asymptotes of hyperbola
- Finding the range of product of two distances.
- Constrain coordinates of a point into a circle
- Position of point with respect to hyperbola
- Length of Shadow from a lamp?
- Show that the asymptotes of an hyperbola are its tangents at infinity points
Related Questions in HOMOTOPY-THEORY
- how to prove this homotopic problem
- Are $[0,1]$ and $(0,1)$ homotopy equivalent?
- two maps are not homotopic equivalent
- the quotien space of $ S^1\times S^1$
- Can $X=SO(n)\setminus\{I_n\}$be homeomorphic to or homotopic equivalent to product of spheres?
- Why do $S^1 \wedge - $ and $Maps(S^1,-)$ form a Quillen adjunction?
- Is $S^{n-1}$ a deformation retract of $S^{n}$ \ {$k$ points}?
- Connection between Mayer-Vietoris and higher dimensional Seifert-Van Kampen Theorems
- Why is the number of exotic spheres equivalent to $S^7,S^{11},S^{15},S^{27}$ equal to perfect numbers?
- Are the maps homotopic?
Related Questions in HOMOLOGY-COHOMOLOGY
- Are these cycles boundaries?
- Cohomology groups of a torus minus a finite number of disjoint open disks
- $f$ - odd implies $d(f)$ - odd, question to the proof
- Poincarè duals in complex projective space and homotopy
- understanding proof of excision theorem
- proof of excision theorem: commutativity of a diagram
- exact sequence of reduced homology groups
- Doubts about computation of the homology of $\Bbb RP^2$ in Vick's *Homology Theory*
- the quotien space of $ S^1\times S^1$
- Rational points on conics over fields of dimension 1
Related Questions in SIMPLICIAL-STUFF
- Homotopy type of simplicial complexes
- $\Pi_f$ for a morphism $f$ between simplicial sets
- Trouble with changing the fractions into matrix-vector from
- A fibration $p : E \rightarrow X$ is trivial iff $p$ is a homotopy equivalence
- Quillen equivalence between sSet (Joyal's model structure) and sSetCat (Bergner's one)
- The choice of cofibrant approximation is contractible
- Do Homotopy limits commute with right Quillen functors
- Necessary conditions for a Reedy fibrant diagram
- $\infty$-categories definition disambiguation
- For simplicial abelian group quotient map to the quotient chain complex is homotopy equivalence.
Trending Questions
- Induction on the number of equations
- How to convince a math teacher of this simple and obvious fact?
- Find $E[XY|Y+Z=1 ]$
- Refuting the Anti-Cantor Cranks
- What are imaginary numbers?
- Determine the adjoint of $\tilde Q(x)$ for $\tilde Q(x)u:=(Qu)(x)$ where $Q:U→L^2(Ω,ℝ^d$ is a Hilbert-Schmidt operator and $U$ is a Hilbert space
- Why does this innovative method of subtraction from a third grader always work?
- How do we know that the number $1$ is not equal to the number $-1$?
- What are the Implications of having VΩ as a model for a theory?
- Defining a Galois Field based on primitive element versus polynomial?
- Can't find the relationship between two columns of numbers. Please Help
- Is computer science a branch of mathematics?
- Is there a bijection of $\mathbb{R}^n$ with itself such that the forward map is connected but the inverse is not?
- Identification of a quadrilateral as a trapezoid, rectangle, or square
- Generator of inertia group in function field extension
Popular # Hahtags
second-order-logic
numerical-methods
puzzle
logic
probability
number-theory
winding-number
real-analysis
integration
calculus
complex-analysis
sequences-and-series
proof-writing
set-theory
functions
homotopy-theory
elementary-number-theory
ordinary-differential-equations
circles
derivatives
game-theory
definite-integrals
elementary-set-theory
limits
multivariable-calculus
geometry
algebraic-number-theory
proof-verification
partial-derivative
algebra-precalculus
Popular Questions
- What is the integral of 1/x?
- How many squares actually ARE in this picture? Is this a trick question with no right answer?
- Is a matrix multiplied with its transpose something special?
- What is the difference between independent and mutually exclusive events?
- Visually stunning math concepts which are easy to explain
- taylor series of $\ln(1+x)$?
- How to tell if a set of vectors spans a space?
- Calculus question taking derivative to find horizontal tangent line
- How to determine if a function is one-to-one?
- Determine if vectors are linearly independent
- What does it mean to have a determinant equal to zero?
- Is this Batman equation for real?
- How to find perpendicular vector to another vector?
- How to find mean and median from histogram
- How many sides does a circle have?
Intuitively you can think about the quotient space of such a good CW pair $(X,Y)$, as collapsing all cells in $Y$ to a point. So if you collapse $Y$ you get all cells of $X$ which were not in $Y$ but glued in maybe a very different way than before. So the dimension (i.e. the highest dimension where there exists a cell) of the quotient space decreases at most.
So for a more rigorous attempt, note that all interiors of cells in $X/Y$ were already interiors of cells in $X$. So we conclude: on every cell which is not contained in $Y$ (those don't interest us, since they are collapsed to one single point) the quotient maps restricts to a homeomorphism on the interior. This determines the dimension of a cell: the dimension of the quotient CW space is bounded by the dimension of the original CW space.
Hope this helps