I want to prove that the suspension $\Sigma X$ of a CW-complex $X$ is a CW-complex, buy I'm starting with CW-complexes and I don't have a clue of how start, so I'd appreciate any help. Thanks.
$$\Sigma X=\frac{X \times I}{X \times \{0,1\}}$$
2026-03-25 17:45:20.1774460720
Suspension of a CW complex
1.6k Views Asked by Bumbble Comm https://math.techqa.club/user/bumbble-comm/detail At
1
There are 1 best solutions below
Related Questions in ALGEBRAIC-TOPOLOGY
- How to compute homology group of $S^1 \times S^n$
- the degree of a map from $S^2$ to $S^2$
- Show $f$ and $g$ are both homeomorphism mapping of $T^2$ but $f$ is not homotopy equivalent with $g.$
- Chain homotopy on linear chains: confusion from Hatcher's book
- Compute Thom and Euler class
- Are these cycles boundaries?
- a problem related with path lifting property
- Bott and Tu exercise 6.5 - Reducing the structure group of a vector bundle to $O(n)$
- Cohomology groups of a torus minus a finite number of disjoint open disks
- CW-structure on $S^n$ and orientations
Related Questions in CW-COMPLEXES
- CW complexes are compactly generated
- If $(X, \Gamma)$ is a cell complex and $e \in \Gamma$, then the image of a characteristic map for $e$ equals its closure
- Universal covering space of $\mathbb{R^3}\setminus S^1$
- Are all cell decompositions useful?
- Give an example of an open cell in a cell complex for which the image of the characteristic map is not a closed cell.
- Computing cellular homology of the sphere using a different CW structure
- Showing that a CW complex is compactly generated
- CW complex with fundamental group $\Bbb Z/n$
- CW complex definition ambiguity
- CW complex for Möbius strip
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?
It is a general result (that I invite you to prove) that the product of CW-complexes (one of which is finite) is still a CW-complex, using the product of the cells involved. Hence $X \times I$ is a CW-complex. Moreover, $X \times \{0,1\}$ is a subcomplex thereof, so that $(X \times I,X \times \{0,1\})$ is a relative subcomplex (see this http://www.math.uchicago.edu/~may/CONCISE/ConciseRevised.pdf , at page 73-74): this implies that $\Sigma X=(X\times I) /(X \times \{0,1\})$ is a CW-complex.