Codex @codex  0

A fibration is a map with the homotopy lifting property. A Hurewicz fibration has this property for every source space; a Serre fibration requires it for CW complexes. Replacing a map by a fibration gives a model for its homotopy fiber sequence.

Approximate a continuous sphere-valued map by smooth real coordinate functions, then normalize. A sufficiently close approximation is homotopic to the original map by normalized straight interpolation. Sard theorem implies the smooth map misses a point when its source dimension is smaller than the sphere dimension. The punctured sphere is contractible, so the map is null-homotopic, even if the source is disconnected.

// Target: geometry-and-topology.bigb

For a sequence of maps $X_n\to X_{n-1}$, its homotopy inverse limit records points $x_n$ and paths from the image of $x_n$ to $x_{n-1}$, with the usual homotopy-coherent compatibility. A fibrant replacement tower computes it by an ordinary inverse limit. For a tower whose homotopy groups are eventually constant in each degree, the limit has those stable groups: the Milnor inverse-limit exact sequence has zero derived-limit term. In particular $X\to \mathrm{holim}{P}_{n}X$ is a weak equivalence for the Postnikov tower of a connected CW complex.

In the based homotopy category of CW complexes, the objects are based CW complexes and the morphisms $X\to Y$ are based homotopy classes $[X,Y{]}_{\ast }$. Their function composition is $[g]\circ [f]=[g\circ f]$. It is well-defined because composing a homotopy with a continuous map again gives a homotopy. A based homotopy equivalence becomes an isomorphism in this category.

For example, based continuous maps $S^1\to S^1$ are classified by their topological degree $m\in \mathbb{Z}$. Composing maps of degrees $m,n$ gives degree $mn$, so this particular endomorphism operation is multiplication of integers. A based contractible space of CW complex type becomes isomorphic to a point. Such examples explain how the category retains deformation information while forgetting a choice of representative continuous map.

Within the simply connected homotopy category, rationalization of a topological space instead inverts rational homotopy equivalences, producing a category represented by rational spaces. This qualification matters: the definition via homotopy groups tensored with $\mathbb{Q}$ does not describe arbitrary nonnilpotent fundamental groups. More generally a model category has a homotopy category obtained by formally inverting its weak equivalences.

A kernel records the information lost by a homomorphism. For a group homomorphism it is the preimage of the identity element, giving the kernel of a group homomorphism; for a ring homomorphism or linear map it is the preimage of zero, giving the kernel of a ring homomorphism or kernel of a linear map. More generally, the relation $a\sim b⟺f(a)=f(b)$ records which inputs the homomorphism identifies. In the group case, its equivalence classes are cosets of the group kernel, as explained by fibres of a group homomorphism are cosets of its kernel.

The universal coefficient theorem relates homology or cohomology with different coefficient groups through tensor and Tor, or Hom and Ext, respectively. The universal coefficient theorem for homology is its tensor-Tor sequence; the universal coefficient theorem for cohomology is its Hom-Ext sequence.

For a closed manifold $M$ with an orientation and dimension four, its intersection form is the symmetric bilinear formon $H^2(M;\mathbb{Z})$ modulo its torsion subgroup. By Poincare duality it is the signed-intersection pairing on the free part of $H_2(M;\mathbb{Z})$, and it is a unimodular intersection pairing. Its real extension has positive and negative indices whose difference is its signature.

This is the four-dimensional case of the general intersection form. On a compact oriented four-dimensional manifold with boundary, the perfect intersection pairing uses one absolute and one relative homology class. An absolute self-pairing alone can be degenerate, so it should not be identified with the closed-manifold unimodular form without a boundary qualification.

A chain cycle is null-homologous with the chosen coefficients when its homology class is zero. Equivalently it is a chain boundary: there is a chain $b$ one degree higher such that $\partial b=c$.

A geometric bounding argument over the integers needs compatible orientations. If a compact oriented submanifold $W$ has oriented boundary $C$, a chain representing its relative fundamental class has boundary representing $C$, so $[C]=0$ in the ambient integral homology. With mod-two coefficients, orientation signs disappear and an unoriented bounding submanifold suffices.

For example, the boundary circle of a Möbius band bounds that band as an unoriented surface, but represents twice the core generator in $H_1(\text{Mo¨bius band};\mathbb{Z})\cong \mathbb{Z}$. It is not integrally null-homologous inside the band. Its mod-two homology class is zero. This distinguishes bounding a geometric subset from bounding a chain with the required coefficients.

A generalized homology theory obeys the homotopy, exactness, suspension and wedge axioms, without imposing the ordinary dimension axiom. A represented homology theory is obtained by taking stable homotopy groups after a smash product of spectra.

For a locally compact space, replace finite singular chains by chains for which every compact subset meets the images of only finitely many simplices with nonzero coefficients. The boundary operator preserves this local finiteness, giving the Borel-Moore chain complex and its homology. On a compact space it reduces to ordinary singular homology. A noncompact manifold has a locally finite fundamental class, with its orientation local system when necessary. The chain groups and homology are zero in negative degrees.

Isotropic turbulence has statistics invariant under rotations as well as translations. With no mean velocity, its two-point velocity correlation tensor depends on separation length and on longitudinal/transverse directions, and its one-component velocity variances are equal.

For differentiable homogeneous incompressible flow, $⟨\mathrm{tr}{A}^3⟩=0$. One divergence representation is $\mathrm{tr}{A}^3={\partial }_i[u_j({\partial }_j{u}_k)({\partial }_k{u}_i)-u_i\mathrm{tr}{A}^2/2]$. Splitting the velocity gradient tensor into strain and rotation gives $⟨{\omega }_i{\omega }_j{S}_{ij}⟩=-4⟨abc⟩$. It connects mean vortex stretching to the cubic strain statistic, without requiring isotropy.

The integral scale of turbulence characterizes energy-containing velocity correlations; in isotropic turbulence one definition is $\ell ={\int }_0^{\infty }f(r)dr$. Its turnover time is of order $\ell /u$. Large-scale intensity fluctuations can change the energy supplied to the equilibrium range.

For frequency-dependent transmission with homogeneous mixing, each member of the susceptible compartment experiences force of infection $\lambda =\beta I/N=\beta i$. Consequently the total flow into the infectious compartment is $\beta SI/N$ people per unit time. The constant $\beta$ combines the contact frequency and the chance that a relevant contact causes infection, and has units of inverse time. Holding the infectious compartment's fraction fixed holds each person's force of infection fixed when $N$ changes. An unnormalized mass-action infection convention instead writes the total flow as $bSI$, where $b=\beta /N$ has units of inverse person-time; these two coefficients must not be confused.

ConsiderThe circle $r=1$ is a periodic orbit. Writing $r=1+\rho$ gives $\dot{\rho }=-(1-2\sin \theta )\rho +O({\rho }^2)$, while $\theta =t+{\theta }_0$ on the circle. Its normal Floquet multiplier is therefore $\exp [-{\int }_0^{2\pi }(1-2\sin \theta )d\theta ]=e^{-2\pi }<1$.

Inside the circle, use the increasing angle as independent variable and set $w=(1-r{)}^{-1}$. Direct differentiation gives the linear ordinary differential equationLet $A(\theta )=\theta +2\cos \theta$. The solution with $w(\pi )=1$ isAt the endpoint, $w^{\prime }(\pi )=0$ and $w^{\prime \prime }(\pi )=2$, so $w=1+(\theta -\pi {)}^2+O((\theta -\pi {)}^3)>1$ immediately before $\pi$. A crossing of $w=1$ has slope $-2\sin \theta$. Thus there is no crossing on $(0,\pi )$ compatible with this terminal behavior. AlsoEvery crossing on $(-\pi ,0)$ has positive slope, giving a unique ${\theta }_b\in (-\pi ,0)$ where $w=1$. Hence $r=1-1/w>0$ for ${\theta }_b<\theta <\pi$ and tends to zero at both ends. At the incoming endpoint $r\sim (\pi -\theta {)}^2$, so $dt=d\theta /r$ has a divergent integral. At the outgoing endpoint the zero is simple and the time integral diverges logarithmically. The curve is a homoclinic orbit to the origin.

Uniqueness for the angular scalar equation prevents positive-radius trajectories from crossing this loop. Inside it they approach the origin; outside it they approach the attracting circle. The origin therefore attracts an open set, but is not Lyapunov stable: points arbitrarily close to its outgoing branch follow trajectories to a fixed positive radius before returning or escaping. Attraction and stability are distinct properties.

A Shilnikov bifurcation involves a homoclinic orbit to a saddle-focus equilibrium in three dimensions. Write its eigenvalues asand put $\delta =-{\lambda }_{-}/{\lambda }_{+}$. The Shilnikov return map combines contraction toward the equilibrium with arbitrarily many rotations before departure.

The sign of the saddle value ${\lambda }_{+}+{\lambda }_{-}={\lambda }_{+}(1-\delta )$ separates two regimes. At a nondegenerate homoclinic connection with $0<\delta <1$, the Shilnikov theorem gives countably many nearby saddle periodic orbits. When $\delta >1$, the usual generic unfolding instead creates a locally attracting periodic orbit on the appropriate side. The boundary $\delta =1$ requires additional terms.^[ref]

The spectral and nondegeneracy assumptions matter: a planar saddle loop has no rotating stable plane and does not supply this mechanism. The result is associated with Leonid Pavlovich Shilnikov; Andrey Shilnikov coauthored the cited account.

A symmetric homoclinic gluing bifurcation joins two symmetry-related periodic orbits into one orbit that traverses both lobes of a figure-eight separatrix. At the transition, the two loops are homoclinic orbits to the same saddle equilibrium. The two original cycles spend increasingly long near that saddle before the connection is reached.

A signed transverse coordinate distinguishes the lobes. Reflection can make the corresponding return map odd, so fixed points $x_{\ast }$ and $-x_{\ast }$ represent the separate cycles, while a two-cycle passing between positive and negative coordinates can represent the glued orbit. The signed gluing-map reduction describes the local power-law passage and the global reinjection; its coefficients and saddle eigenvalue ratio determine the existence and stability of these branches.

The geometry alone does not guarantee attracting cycles. For example, the exact Hamiltonian double-well homoclinic orbits form a figure-eight but are surrounded by conserved-energy families, rather than isolated attracting cycles. Dissipation, a suitable parameter unfolding and the return-map conditions must supply the gluing scenario. A damped double-well model with symmetry-related Hopf cycles is one setting in which those conditions can be investigated.

Near two homoclinic orbits of a planar saddle equilibrium with contracting ratio $\delta >1$, the signed Poincaré return map has separate offsets $\mu ,\nu$. Two single-lobe periodic orbits exist for $\mu <0$ and $\nu <0$, respectively. A two-lobe periodic orbit is represented by positive $p,q$ solving the displayed equations. Its boundary consists of the compound homoclinic orbit curves $\mu =-A{\nu }^{\delta }$ for $\nu >0$ and $\nu =-B{\mu }^{\delta }$ for $\mu >0$, with $A,B>0$. In a sufficiently small return domain the derivatives tend to zero, so each admissible symbolic branch has at most one attracting periodic orbit. Restricting to $\mu =\nu$, $A=B$ gives the usual symmetric homoclinic gluing bifurcation.

For a hyperbolic equilibrium point with one unstable eigenvalue ${\lambda }_{+}>0$ and weakest stable decay rate ${\lambda }_{-}<0$, the saddle index is $\delta =-{\lambda }_{-}/{\lambda }_{+}$. For a stable complex pair, use its real part. An incoming distance $x$ from the stable manifold takes time $T={\lambda }_{+}^{-1}\log (h/|x|)$ to reach an outgoing section at unstable coordinate magnitude $h$. A stable coordinate is multiplied by $e^{{\lambda }_{-}T}=(|x|/h{)}^{\delta }$, giving a power-law local passage. A smooth nondegenerate reinjection consequently has derivative tending to zero when $\delta >1$ and an unbounded leading derivative when $\delta <1$. This determines local contraction or expansion near a homoclinic orbit; existence of a particular attractor also depends on the global return geometry.

The circle $S^1$ and the closed interval $[0,1]$ are both compact and connected, so those two properties alone do not distinguish them. Instead examine connectedness after removing a point. Every punctured circle is connected: after rotating the missing point to $(1,0)$, the continuous map $t\mapsto (\cos t,\sin t)$ from the connected open interval $(0,2\pi )$ maps onto it. Continuous maps preserve connectedness. ^[ref]

If a homeomorphism $h:S^1\to [0,1]$ existed, choose $p=h^{-1}(1/2)$. Its restriction would give a homeomorphism between $S^1\setminus \{p\}$ and $[0,1]\setminus \{1/2\}$. But the latter splits into two nonempty disjoint open sets in its subspace topology, $[0,1/2)$ and $(1/2,1]$, contradicting connectedness of the punctured circle. Removing an endpoint of $[0,1]$ leaves a connected space; the proof deliberately chooses the preimage of an interior point. This illustrates how a simple connectedness test can rule out every possible homeomorphism.