- Paths and endpoint-fixed homotopy
- Loops form the fundamental group
- Distinct points form configuration spaces
- Lift a loop to labelled trajectories
- Braids, pure braids, and endpoint permutations
A braid records the motion of distinct points. The right space therefore has a point for every possible configuration, and a braid becomes a loop in that space. Deforming the braid becomes deforming the loop. We build this definition from paths, then explain exactly why forgetting labels produces the full braid group.
3.1. Paths and the fundamental group
Definition 3.1. Paths and based loops
Write \(I=[0,1]\). A path in a topological space \(X\) is a continuous map \(\alpha:I\to X\). Its endpoints are \(\alpha(0)\) and \(\alpha(1)\). A loop based at \(x_0\) has both endpoints equal to \(x_0\).
Definition 3.2. Homotopy relative to a subspace
A homotopy from a continuous map \(f:Y\to X\) to \(g:Y\to X\) is a continuous \(H:Y\times I\to X\) with \(H(y,0)=f(y)\) and \(H(y,1)=g(y)\). It is relative to \(A\subseteq Y\) when \(H(a,s)=f(a)=g(a)\) for every \(a\in A\). Path homotopy is the case \(Y=I\), \(A=\{0,1\}\).
Two paths \(\alpha,\beta\) with the same endpoints are homotopic with endpoints fixed if there is a continuous map \(H:I\times I\to X\) such that
\[ H(t,0)=\alpha(t),\quad H(t,1)=\beta(t),\qquad H(0,s)=\alpha(0),\quad H(1,s)=\alpha(1). \]The parameter \(t\) runs along a path; \(s\) runs through the deformation. For based loops the entire left and right edges of this square stay at \(x_0\). Write \([\alpha]\) for the endpoint-fixed homotopy class. Fixing endpoints matters: it keeps the starting configuration of a braid unchanged throughout a deformation.
Definition 3.3. Fundamental group
The fundamental group \(\pi_1(X,x_0)\) is the set of these based-loop classes. Multiply by following the first loop and then the second:
\[ (\alpha*\beta)(t)= \begin{cases}\alpha(2t), & 0\le t\le\tfrac12,\\ \beta(2t-1), & \tfrac12\le t\le1. \end{cases} \qquad [\alpha][\beta]=[\alpha*\beta]. \]Prerequisite: pasting and Lebesgue numbers
Lemma 3.A. Compactness tools
A finite closed cover permits pasting continuous maps agreeing on overlaps. Every open cover of a compact metric space has a positive Lebesgue number.
Proof.
The finite closed pasting lemma says that continuous maps on finitely many closed subsets covering a space combine to a continuous map if they agree on overlaps. For a closed set in the target, its inverse image is a finite union of sets closed in those closed subsets, hence closed in the whole domain. This proves the lemma and applies to interval subdivisions and rectangular grids.
The Lebesgue-number lemma says that an open cover of a compact metric space has a number \(\delta>0\) such that every subset of diameter less than \(\delta\) lies in one cover member. To prove it, choose for each point \(x\) a radius \(r_x>0\) with the ball \(B(x,2r_x)\) contained in a cover member. Finitely many balls \(B(x,r_x)\) cover the space. Take \(\delta\) to be the minimum of those finitely many radii. A nonempty set of diameter less than \(\delta\), containing a point of one of these smaller balls, lies in the corresponding larger ball by the triangle inequality. Thus sufficiently short interval pieces or sufficiently small grid cells fit into single cover members.
□Proposition 3.4. The fundamental-group laws
Endpoint-fixed homotopy is an equivalence relation. Concatenation induces a group law on based-loop classes, with constant-loop identity and reversed-path inverse.
Proof of Proposition 3.4
Proof.
The two pieces agree at the join, so the pasting lemma makes this path continuous. The identity is the constant loop \(c_{x_0}(t)=x_0\), and the inverse is \(\bar\alpha(t)=\alpha(1-t)\). These operations satisfy the group laws on homotopy classes; the justification is below. See also Alphabeta Math's definition and group-law proof.
Endpoint-fixed homotopy is an equivalence relation: the constant deformation proves reflexivity, replacing \(s\) by \(1-s\) proves symmetry, and concatenating two deformations in their \(s\)-coordinate proves transitivity. The pasting lemma applies because they agree along the seam. Concatenating two endpoint-fixed homotopies in their \(t\)-coordinate shows that the product does not depend on representatives.
Changing speed does not change a path class. If \(r:I\to I\) is continuous with \(r(0)=0,r(1)=1\), then \(\alpha((1-s)t+sr(t))\) deforms \(\alpha\) into \(\alpha\circ r\), fixing endpoints. The two bracketings of three loops have the same successive segments but different time allocations. Interpolating their piecewise-linear time schedules gives their homotopy, proving associativity. The same argument removes the pause in \(c*\alpha\) or \(\alpha*c\), proving the identity laws.
For cancellation set \(f(t)=2t\) for \(t\le1/2\) and \(f(t)=2-2t\) for \(t\ge1/2\). The formula \(H(t,s)=\alpha((1-s)f(t))\) contracts \(\alpha*\bar\alpha\) to the constant loop at \(\alpha(0)\), keeping endpoints fixed. Apply the same formula to \(\bar\alpha\) for the other inverse law. These explicit deformations prove all group axioms.
□Prerequisite: induced maps and change of basepoint
Proposition 3.B. Functoriality and basepoint transport
A based continuous map induces a homomorphism of fundamental groups. A path \(u:x_0\to x_1\) induces an isomorphism \([\alpha]\mapsto[\bar u*\alpha*u]\).
Proof.
A continuous map \(f:X\to Y\) with \(f(x_0)=y_0\) induces \(f_*:\pi_1(X,x_0)\to\pi_1(Y,y_0)\), \([\alpha]\mapsto[f\circ\alpha]\). Composing homotopies with \(f\) proves well-definedness, and composing concatenated loops proves that it is a homomorphism. A homeomorphism gives an isomorphism, since its inverse induces the inverse map.
If \(u\) is a path from \(x_0\) to \(x_1\), the map \([\alpha]\mapsto[\bar u*\alpha*u]\) is an isomorphism from \(\pi_1(X,x_0)\) to \(\pi_1(X,x_1)\). Adjacent \(u*\bar u\) segments cancel by the contraction above; reversing \(u\) gives the inverse. A different choice of connecting path may change this isomorphism by conjugation. Thus a path-connected space has isomorphic fundamental groups at different basepoints, but no preferred identification without a choice of path. Alphabeta Math proves this basepoint-change statement.
Only products, subspaces, quotient topology, and the lifting argument below are needed here. Higher homotopy groups, covering classification, and homology are not prerequisites for the definition.
□3.2. Ordered and unordered configurations
Fix \(n\ge3\). Let \(D=\{z\in\mathbb C:|z|\le1\}\) be the closed unit disk and \(D^\circ=\{z:|z|<1\}\) its interior. Choose the ordered base configuration \(q=(q_1,\ldots,q_n)\), with \(q_j=-1+2j/(n+1)\), and its underlying set \(Q=\{q_1,\ldots,q_n\}\).
Definition 3.5. Configuration spaces
For a space \(X\), its ordered configuration space is
\[ F_n(X)=\{(z_1,\ldots,z_n)\in X^n:z_i\ne z_j\text{ whenever }i\ne j\}, \]Give \(F_n(X)\) the subspace topology of \(X^n\). Let \(S_n\) act by
\[ (\tau\cdot z)_j=z_{\tau^{-1}(j)}. \]The unordered configuration space is the orbit space with its quotient topology:
\[ C_n(X)=F_n(X)/S_n. \]An orbit is precisely an \(n\)-element subset of \(X\). Let \(p:F_n(X)\to C_n(X)\) forget the ordering. The quotient topology means that \(V\subset C_n(X)\) is open exactly when \(p^{-1}(V)\) is open in \(F_n(X)\). This topology, rather than merely the set of subsets, makes continuity of unordered motion meaningful. These are the ordered and unordered definitions used by Alphabeta Math.
Alphabeta Math: ordered configurations · unordered configurations
Worked example for Definition 3.5
Counterexample: A real-coordinate crossing is not a collision
Counterexample: Ordered configurations on a line need not be path connected
Lemma 3.6. Free permutation action
For \(X=D^\circ\), the space \(F_n(X)\) is open in \(\mathbb R^{2n}\), and the displayed permutation action is a free action by homeomorphisms.
Counterexample: Ordered configurations on a line need not be path connected
Proof.
A coordinate is a label: the first point remains point 1 as it moves. When \(X=D^\circ\), this is an open subset of \(\mathbb R^{2n}\): at any tuple, the positive minimum separation and positive distances from the boundary allow sufficiently small independent movements without collisions or escape.
The symmetric group \(S_n\), the group of permutations of \(\{1,\ldots,n\}\), acts by
Coordinate permutations are homeomorphisms, and this is a left action because \((\tau\upsilon)^{-1}=\upsilon^{-1}\tau^{-1}\). It is free: if \(\tau\cdot z=z\), distinct coordinates force \(\tau(j)=j\) for every \(j\).
□Definition 3.7. Covering map
A covering map is a map for which every point downstairs has an open neighbourhood whose inverse image is a disjoint union of open sets, each mapped homeomorphically onto that neighbourhood.
Theorem 3.8. The configuration covering
The forgetful map \(p:F_n(D^\circ)\to C_n(D^\circ)\) is an \(n!\)-sheeted covering.
Proof.
Here is the local model for \(p\). Around the distinct points of a configuration choose pairwise disjoint small open disks \(U_1,\ldots,U_n\subset D^\circ\). Let \(V\) consist of configurations with exactly one point in each \(U_j\). Then
\[ p^{-1}(V)=\coprod_{\tau\in S_n}\tau\cdot(U_1\times\cdots\times U_n). \]These \(n!\) products are disjoint open sets. On each, forgetting labels is bijective onto \(V\). It is a homeomorphism: the quotient map is open because for any open \(O\), \(p^{-1}(p(O))=\bigcup_\tau\tau\cdot O\) is open. This proves that \(p\) is an \(n!\)-sheeted covering, not just a many-to-one map. Compare the disjoint-neighbourhood proof.
□Consequently, an unordered path and an initial ordering determine unique continuous labelled trajectories. Locally, the point in each small disk inherits its label; when moving to the next neighbourhood, the existing labels determine which ordering to use.
Path and homotopy lifting
Theorem 3.C. Lifting and injectivity
A path and its initial ordering determine a unique lift. Homotopies lift relative to a chosen initial lift. In particular, \(p_*\) is injective on fundamental groups.
Alphabeta Math: path lifting · homotopy lifting · injectivity
Proof.
For a path, pull back evenly covered neighbourhoods to an open cover of \(I\). Compactness and the Lebesgue-number lemma give a finite subdivision so each subinterval maps into one neighbourhood. Choose its sheet using the initial value and continue across successive joins. Local inverse maps give continuity. Two lifts agreeing at one time agree near that time; the subdivision therefore proves uniqueness.
For a homotopy on \(I^2\), the same compactness argument gives a finite rectangular grid whose cells each map into an evenly covered neighbourhood. Work through the bottom row from left to right, then through successive rows. On a new cell select the sheet containing the already lifted bottom-left corner, and compose the homotopy on the whole cell with that sheet's inverse map. Its bottom edge, and its left edge when already constructed, agree with the existing lift by path uniqueness: both lift the same edge path and agree at the corner. Thus the construction preserves the prescribed bottom lift and all shared edges. The finite closed pasting lemma gives a continuous lift of the square. Uniqueness follows by applying path uniqueness on each vertical interval. This is the square case of the covering homotopy-lifting theorem; see also Hatcher, Algebraic Topology, §1.3.
In particular, the terminal ordering of a lifted based loop is unchanged under an endpoint-fixed homotopy: the terminal edge lifts to a continuous path in the finite discrete fibre, hence is constant. If a loop upstairs projects to a null-homotopic loop, lifting its contraction keeps both side edges at its initial point and the final edge constant. It therefore contracts upstairs too. Thus \(p_*\) is injective. This is the covering injection theorem, with its proof in the present setting.
□3.3. Braids as point motions
Definition 3.9. Full and pure configuration braids
Define
\[ B_n^{\mathrm{conf}}=\pi_1(C_n(D^\circ),Q),\qquad P_n=\pi_1(F_n(D^\circ),q). \]In the full group, a loop returns to the same set \(Q\); its points may exchange places. In the pure braid group, every labelled point returns to its own \(q_j\). The covering injection identifies \(P_n\) with a subgroup of \(B_n^{\mathrm{conf}}\). We retain the superscript until Chapter 5 proves the identification with the presented group from Chapter 2.
Definition 3.10. Geometric braid
A geometric braid based at \(Q\) is the union of the graphs \(\{(z_j(t),t):0\leq t\leq1\}\), where the \(z_j\) are continuous, pairwise distinct at every time, and their initial and terminal sets are \(Q\). A braid isotopy is a continuous family of such motions fixing every strand endpoint.
Alphabeta Math: geometric braids · braid isotopy
Worked example for Definition 3.10
Counterexample: A real-coordinate crossing is not a collision
Proposition 3.11. Slicing and tracing
Tracing a configuration loop and slicing a geometric braid give inverse identifications of their deformation classes, compatible with chronological concatenation.
Proof.
Lift a based unordered loop \(\gamma\) to \((z_1(t),\ldots,z_n(t))\) starting at \(q\). Its trajectories
\[ t\longmapsto(z_j(t),t)\quad\text{in }D^\circ\times I \]are the strands of a geometric braid. Each meets every time slice exactly once. Two strands cannot intersect: an intersection would have the same time coordinate and then \(z_i(t)=z_j(t)\), a forbidden collision. Conversely, slicing a braid given by these time-monotone trajectories gives a configuration loop. An endpoint-fixed configuration homotopy lifts to a continuous family of collision-free strands with their endpoints fixed; slicing such a family reverses the construction. Concatenation corresponds to stacking in chronological order. These inverse constructions identify configuration-loop classes with geometric braids modulo deformation through braids. Compare Alphabeta Math's comparison theorem: its geometric stacking uses the opposite order, so its raw slicing map reverses products. Here we explicitly use chronological stacking. The deformation is through time-monotone braids, rather than arbitrary link isotopy.
□Definition 3.12. Positive elementary exchange
Fix \(1\le i<n\), set \(a=(q_i+q_{i+1})/2\), and \(r=(q_{i+1}-q_i)/2\). Keep the other points fixed and move
\[ z_i(t)=a-r e^{\pi\mathrm{i}t},\qquad z_{i+1}(t)=a+r e^{\pi\mathrm{i}t}. \]Proposition 3.13. Elementary exchanges are configuration loops
The preceding formulas define a counterclockwise loop \(\sigma_i\) in \(C_n(D^\circ)\), whose ordered lift interchanges \(q_i\) and \(q_{i+1}\).
Worked example for Proposition 3.13
Counterexample: A real-coordinate crossing is not a collision
Proof.
Here \(\mathrm{i}^2=-1\); its upright font distinguishes it from the index \(i\). The disk of radius \(r\) about \(a\) lies in \(D^\circ\) and contains no other \(q_j\), while the moving points remain distance \(2r\) apart. Both rotate counterclockwise: the initially left point passes below the real axis, and the initially right point passes above. At \(t=1\) they have exchanged places. Their unordered motion is a loop representing the positive generator \(\sigma_i\); the ordered motion is not a loop.
□Solid tracks show the motion so far; faint tracks show its continuation. Drag to rotate, or use the camera sliders. All sliders support arrow keys. Playback is manual; reduced-motion mode advances one time step per press.
3.4. The permutation exact sequence
Theorem 3.14. Full and pure braids
For the inverse endpoint permutation \(\rho([\gamma])=e_\gamma^{-1}\), the sequence
\[ 1\longrightarrow P_n\xrightarrow{p_*}B_n^{\mathrm{conf}} \xrightarrow{\rho}S_n\longrightarrow1. \]is exact. In particular, \(B_n^{\mathrm{conf}}/P_n\cong S_n\).
Proof.
For a lifted loop let \(e_\gamma(j)=k\) mean that the strand starting at \(q_j\) ends at \(q_k\). This forward endpoint permutation satisfies \(e_{\alpha*\beta}=e_\beta\circ e_\alpha\), because we traverse \(\alpha\) first. With the usual permutation multiplication \(\tau\upsilon=\tau\circ\upsilon\), define instead
\[ \rho([\gamma])=e_\gamma^{-1},\qquad \rho([\alpha][\beta])=\rho([\alpha])\rho([\beta]). \]Thus \(\rho\) records, for each final position, the label that arrived there. Homotopy lifting proves that it depends only on \([\gamma]\). Each elementary exchange maps to the adjacent transposition \((i\ i+1)\). Adjacent transpositions generate \(S_n\): successively swapping adjacent out-of-order entries sorts any list into \(1,\ldots,n\). Hence \(\rho\) is surjective.
A loop lies in \(\ker\rho\) exactly when its lift ends at \(q\), making it an ordered loop. Combined with the injectivity of \(p_*\), this proves the short exact sequence
“Exact” here says that the first map is injective, its image is precisely the kernel of the second, and the second is onto. In particular \(P_n\) is normal and \(B_n^{\mathrm{conf}}/P_n\cong S_n\): two braids give the same permutation exactly when their quotient is pure. Explicitly, the map \(gP_n\mapsto\rho(g)\) is well defined and injective because \(\ker\rho=P_n\), is a homomorphism by the quotient multiplication, and is surjective because \(\rho\) is. This is the configuration braid exact sequence, with our inverse-endpoint convention made explicit.
□Example: a nontrivial pure braid
Proposition 3.15. Two elementary exchanges
The element \([\sigma_i]^2\) is pure and nontrivial.
Proof.
The square \(\sigma_i^2\) lifts to a closed ordered loop: its two moving points rotate a full turn. It is nontrivial. Indeed the continuous map \(F_n(D^\circ)\to S^1\) given by \(z\mapsto(z_{i+1}-z_i)/|z_{i+1}-z_i|\) sends this loop to \(t\mapsto e^{2\pi\mathrm{i}t}\). That circle loop cannot contract: under the covering \(\mathbb R\to S^1\), \(u\mapsto e^{2\pi\mathrm{i}u}\), its lift starting at 0 ends at 1, whereas a constant loop ends at 0; lifted endpoints are homotopy-invariant. This map is a covering because any sufficiently short open arc has a continuous angle coordinate, and its inverse image consists of integer translates of that coordinate interval, each mapped homeomorphically to the arc. A contraction in ordered configuration space would therefore give an impossible contraction on the circle. Covering injectivity proves nontriviality in the full group too. The endpoint permutation detects purity, not all braiding.
□3.5. Disk and plane models
Theorem 3.16. Transport to the plane
The map \(h(z)=z/(1-|z|)\) is a homeomorphism \(D^\circ\to\mathbb C\), and it induces
\[ \pi_1(C_n(D^\circ),Q)\cong\pi_1(C_n(\mathbb C),h(Q)). \]Proof.
The open disk and the plane are different spaces with an explicit homeomorphism:
\[ h:D^\circ\longrightarrow\mathbb C,\qquad h(z)=\frac{z}{1-|z|}, \qquad h^{-1}(w)=\frac{w}{1+|w|}. \]Both maps are continuous and inverse: the radius \(r\) becomes \(r/(1-r)\), whose inverse is \(R/(1+R)\), while the angle stays fixed. Applying \(h\) to every coordinate preserves distinctness and commutes with label permutations. It therefore gives homeomorphisms of ordered spaces and, by the quotient topology, of unordered spaces. Consequently
This is a transported-basepoint isomorphism, not equality of spaces. Other chosen planar base configurations give isomorphic groups after choosing connecting paths. Such paths exist: move labelled points one at a time through a finite set of unused staging positions, avoiding the finitely many stationary points. The plane minus finitely many points is path-connected, since a polygonal segment can detour around each forbidden point. Choose staging positions disjoint from both configurations, then move from staging to the target; applying \(h^{-1}\) gives the same fact in the open disk.
□The closed disk and basepoint transport
Proposition 3.D. Interior and closed-disk configurations
Interior inclusion induces fundamental-group isomorphisms for ordered and unordered configurations, with basepoint transport as specified below.
Proof.
The closed disk itself is not homeomorphic to the plane: it is compact and the plane is not. Nevertheless its configuration spaces have the same homotopy type as the interior versions. A homotopy equivalence is a continuous map with a continuous inverse up to homotopy: both composites are homotopic to the appropriate identity. For inclusion \(j:D^\circ\hookrightarrow D\) and \(s:D\to D^\circ\), \(s(z)=z/2\), the induced configuration maps have this property. Coordinatewise radial scalings \(T_u(z)=(1-u/2)z\), \(0\le u\le1\), give homotopies from the identities to the composites. All scaling factors are positive, so no distinct points collide.
The homotopies commute with permutations and descend continuously to unordered spaces. To check joint continuity, the map \(p\times\mathrm{id}_I\) is an open surjection: it takes each basic open product \(O\times J\) to \(p(O)\times J\), which is open since \(p\) is open. An open surjection is a quotient map. The continuous equivariant homotopy upstairs therefore defines a continuous homotopy downstairs by the definition of quotient topology.
Here the fundamental-group isomorphism can be proved directly at the same base configuration. In either the ordered or unordered model, write \(b=q\) or \(b=Q\), respectively, and let \(c(u)=T_u(b)\), a path from \(b\) to \(b/2\) lying in the interior configuration space. For any loop \(\gamma\) at \(b\) in the closed model, its scaling homotopy has side edges \(c\). The two boundary routes of its parameter square give \(\gamma*c\simeq c*(s\circ\gamma)\) with endpoints fixed: interpolate linearly between the bottom-then-right and left-then-top routes in the square and compose with the homotopy. Cancelling \(c*\bar c\) gives \(\gamma\simeq c*(s\circ\gamma)*\bar c\), an interior loop. Thus inclusion is surjective on \(\pi_1\). If an interior loop contracts in the closed model, applying \(s\) gives an interior contraction of its scaled loop. The same square-route argument and cancellation recover an interior contraction of the original loop. Thus inclusion is injective as well. See the closed/interior disk comparison. We use interior points throughout to match the punctured-disk model of the next chapter.
□Sources and further reading
Allen Hatcher, Algebraic Topology, Chapter 1, §1.3, Propositions 1.30–1.31 (printed pp. 60–61), gives the covering homotopy-lifting and fundamental-group injection theorems. The square proofs needed here are supplied above.
The directly linked Alphabeta Math items are also collected, with proofs, in The Fundamental Group, Covering Spaces and Lifting, Ordered and Unordered Configuration Spaces, and Braids as Fundamental Groups of Configuration Spaces. Their sign and multiplication conventions should be checked when comparing formulas; this series always traverses the first loop first and uses counterclockwise positive exchanges.
Worked examples
These calculations and proofs use this series’ chronological loop products, inverse-endpoint mapping classes, and counterclockwise positive point exchanges. Related statements link to their examples.
Worked example 1. A based loop, its reverse, and an explicit null homotopy
Applies to: Definition 3.1, Definition 3.2, Definition 3.3, Proposition 3.4.
Proof and calculation.
In \(X=\mathbb C\setminus\{0\}\) at basepoint 1, put \(\alpha(t)=e^{2\pi it}\). It is a loop, while \(u(t)=e^{\pi it}\) is a path from 1 to \(-1\), not a based loop at 1. Let \(r(t)=2t\) for \(t\le1/2\), \(r(t)=2-2t\) afterwards. The loop \(\alpha*\bar\alpha\) is \(\alpha(r(t))\). The formula \(H(t,s)=\alpha((1-s)r(t))\) is continuous, keeps both endpoints 1, starts at this loop, and ends at the constant loop. It explicitly proves the inverse law in this case. A reparametrization of any loop by \(\lambda(t)=t^2\) is endpoint-fixed homotopic to it through \(\alpha((1-s)t+st^2)\). Hence different speeds are different paths as functions but the same fundamental-group element.
Worked example 2. Pasting, a Lebesgue number, and transport of basepoint
Applies to: Lemma 3.A, Proposition 3.B.
Proof and calculation.
The two closed half-intervals paste the preceding formula for \(r\), since both values at \(1/2\) are 1. On \([0,1]\), the relatively open cover \(U=[0,3/4),V=(1/4,1]\) has Lebesgue number \(1/4\): a subset of diameter less than \(1/4\) cannot contain both a point \(\le1/4\) and a point \(\ge3/4\); if it misses the first region it lies in \(V\), and if it misses the second it lies in \(U\). Thus a time subdivision of mesh less than \(1/4\) fits into cover members.
The based map \(f(z)=z^2\) on \(\mathbb C\setminus\{0\}\) sends \(\alpha\) to \(e^{4\pi it}\); its argument increases by \(4\pi\), so winding doubles. It sends a concatenation literally to the concatenation of the images, proving the induced homomorphism in this example. With \(u(t)=e^{\pi it}\) from 1 to \(-1\), the loop \(\bar u*\alpha*u\) at \(-1\) has argument increments \(-\pi,2\pi,\pi\), total \(2\pi\). Transport back inserts \(u*\bar u\), which contracts by the explicit backtracking homotopy, giving the inverse basepoint isomorphism rather than silently changing the basepoint.
Worked example 3. Six sheets above one small configuration neighborhood
Applies to: Definition 3.5, Lemma 3.6, Definition 3.7, Theorem 3.8.
Proof and calculation.
For \(n=3\), take disjoint open disks \(U_j=B(q_j,1/16)\). Each lies inside \(D^\circ\); centers are distance at least \(1/2\), exceeding twice the radius. The ordered product \(U_1\times U_2\times U_3\) is an open subset of \(\mathbb R^6\) containing no collisions. Its six coordinate permutations are disjoint: a coordinate cannot lie in two different \(U_j\). Their union is exactly the preimage of the unordered neighborhood consisting of one point per disk. The quotient map restricts on each product to a bijection with continuous inverse obtained by selecting the point in each indicated disk. This inverse is continuous because the quotient is open: the preimage of the image of an open set is the union of its six open translates. Thus these are six actual covering sheets. A permutation fixing an ordered tuple of distinct entries must fix every index; coordinate permutations and their inverses are linear continuous maps, proving freeness and the homeomorphism assertion here.
Worked example 4. Lift a moving point and a homotopy inside a covering chart
Applies to: Theorem 3.C.
Proof and calculation.
In the preceding neighborhood, let \(z_1(t)=-1/2+(t/64)i\), \(z_2(t)=0\), and \(z_3(t)=1/2\). Each remains in its assigned disk. Forgetting labels gives a path with exactly one lift starting at \(q\): selecting the unique point in each disk forces the displayed tuple. The square \(z_1(s,t)=-1/2+(st/64)i\), with the other entries fixed, gives its homotopy lift, again by selection. More generally a lift of a homotopy of loops starting at this tuple has endpoint in the discrete six-element fiber; continuity along that edge keeps the endpoint constant. If an ordered loop projects to a based null homotopy, the lifted square therefore has its endpoint edge constant, giving a based null homotopy upstairs. This directly explains the injectivity assertion with the same local selection rule used in the example.
Worked example 5. Trace an exchange, then trace its pure square
Applies to: Definition 3.9, Definition 3.10, Proposition 3.11, Definition 3.12, Proposition 3.13, Proposition 3.15.
Proof and calculation.
For \(n=3,i=1\), use \(c=-1/4,r=1/4\) and \(z_1=c-re^{\pi it},z_2=c+re^{\pi it},z_3=1/2\). The two moving points stay distance \(1/2\) apart, and the distance to the third is at least \(1/2\); all moduli are at most \(1/2\). Their graphs \((z_j(t),t)\) are disjoint, since equal space-time points would require equal times and then a forbidden collision. Slicing these graphs at a time recovers precisely the unordered configuration; tracing the slices recovers the graphs, including in a continuous endpoint-fixed family. At the top the first two labels swap, so this is a full braid but not an ordered based loop.
Following two exchanges is represented, after keeping track of labels, by \(z_1=c-re^{2\pi it},z_2=c+re^{2\pi it},z_3=1/2\). It is now an ordered loop, so pure. It is not trivial: the discriminant computation in chapter 1 has winding 2, as the angle makes one full turn here. A null homotopy in configurations would give a null homotopy of the nonzero discriminant loop and force winding zero, contradicting argument-lift invariance. This proves nontriviality in the full group as well as in the ordered subgroup.
Worked example 6. The inverse endpoint permutation in a two-exchange product
Applies to: Theorem 3.14.
Proof and calculation.
Start with ordered labels \((1,2,3)\) from left to right and follow \(\sigma_1\), then \(\sigma_2\). The orders become \((2,1,3)\), then \((2,3,1)\). Thus label 1 ends in slot 3, label 2 in slot 1, and label 3 in slot 2: \(e=(132)\). Its inverse is \(\rho=(123)=(12)(23)\), agreeing with the homomorphism under chronological products. A loop has \(\rho=1\) exactly when its ordered lift ends at the initial tuple, exactly the image of an ordered loop; covering injectivity identifies this with \(P_3\). Elementary exchanges realize adjacent transpositions, which generate all permutations by successively moving each required entry to its slot. This verifies kernel and surjectivity in the exact sequence on concrete motions, not only on abstract symbols.
Worked example 7. Radial transport and the closed-disk basepoint correction
Applies to: Theorem 3.16, Proposition 3.D.
Proof and calculation.
For \(h(z)=z/(1-|z|)\), the proposed inverse is \(k(w)=w/(1+|w|)\). Substitution of radii gives \(|h(z)|=|z|/(1-|z|)\), \(|k(w)|=|w|/(1+|w|)\), and both composites are the identity. The formulas are continuous at zero too; injectivity ensures distinct points remain distinct under either map. Thus applying them to every point gives inverse homeomorphisms of both configuration spaces, and applying them to loops and homotopies gives inverse maps of fundamental groups. The base triple maps to \((-1,0,1)\) in the plane, not the original numerical triple.
For configurations in the closed disk, \(R_s(z)=(1-s/4)z\) scales every pairwise distance by the same positive number. It never creates collisions, and for \(s>0\) lands in the interior. At \(s=1\) the base triple has become \((3/4)q\). The path \(u(s)=(1-s/4)Q\) records this change. For a loop \(\alpha\) based at \(Q\), the shrunken loop \(R_1\alpha\) is based at \((3/4)Q\), so its transport back to \(Q\) is \(u*(R_1\alpha)*\bar u\). To see this represents \(\alpha\), let \(u_s(v)=R_{sv}Q\) and use the family \(u_s*(R_s\alpha)*\bar u_s\). It consists of loops based at \(Q\); at \(s=0\) it is \(\alpha\) with constant pauses and at \(s=1\) it is the transported loop. If \(\alpha\) is already interior, the whole homotopy is interior as well. Thus radial shrinking and inclusion give inverse isomorphisms with this explicit basepoint correction.
Counterexamples and distinctions
Each example refutes the stated stronger claim while retaining the hypotheses explicitly checked in its proof.
Counterexample 1. A real-coordinate crossing is not a collision
Related statements: Definition 3.5, Definition 3.10, Proposition 3.13.
Claim refuted. If two projected strands cross, the configuration has left the collision-free space.
Proof.
At \(t=1/2\) in \(\sigma_1\), both moving points have real coordinate \(-1/4\), but their imaginary coordinates are \(-1/4\) and \(1/4\). Their distance remains \(1/2\). The projection to \((\operatorname{Re}z,t)\) identifies them, while the full points in \(D^\circ\times I\) remain distinct. Omitting the imaginary coordinate from the distinctness test would incorrectly prohibit every elementary braid crossing.
Counterexample 2. Ordered configurations on a line need not be path connected
Related statements: Definition 3.5, Lemma 3.6.
Claim refuted. Ordered configurations on every connected space are path connected.
Proof.
The line \(\mathbb R\) is connected, but in \(F_3(\mathbb R)\), a path from \((-1,0,1)\) to \((0,-1,1)\) would make its continuous first-coordinate difference start negative and end positive. The intermediate value theorem forces a zero, a collision, so no such path exists. Each strict ordering region is convex and hence path connected by straight interpolation; the six orderings give six components. The disk construction uses two real dimensions to move around forbidden points and cannot simply be copied to a line.
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