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Let $\gamma$ be the real tautological line bundle over ${\mathbb{RP}}^2$. A homogeneous polynomial of degree $a$ defines a section of the real line bundleIndeed, on a projective line $\ell \subset {\mathbb{R}}^3$ its restriction satisfies $F(rv)=r^{a}F(v)$, exactly the transformation law for a linear functional on ${\ell }^{\otimes a}$. Its zero set is well defined without treating an odd-degree polynomial as a global real-valued function on projective space.

Put $u=w_1(\gamma )\in H^1({\mathbb{RP}}^2;{\mathbb{F}}_2)$. This is the nonzero generator: the tautological line changes sign along the generating projective loop. Dualization preserves this sign, while tensoring multiplies transition signs, soFor generic coefficients both polynomial sections are transverse to zero and their zero curves meet transversely. This genericity follows from parametric transversality: at each projective point some coordinate is nonzero, and varying the coefficient of its $a$th power varies the section's value freely; the two polynomial coefficient sets vary the two values independently. The common zero set is consequently a zero-dimensional submanifold of a compact space, and is finite.

We justify the needed parity of zeros of a real line-bundle section. Along a closed loop transverse to the zero curve, trivialize the pulled-back line bundle on the cut interval. At a simple zero the real component function changes sign. The product of these sign changes equals the sign of the bundle's transition around the loop, which is the evaluation of its first Stiefel–Whitney class. Therefore the intersection parity of the zero curve with every loop equals the evaluation of $w_1$ on that loop. This says that its mod-two Poincare dual is $w_1$, so

The mod-two Poincare duality intersection rule givesThere are no orientation signs in these coefficients, so every transverse intersection contributes one. Two distinct projective lines meet in exactly one point; both have dual class $u$ by the preceding degree-one calculation. Hence $⟨u^2,[{\mathbb{RP}}^2{]}_2⟩=1$, andThis proves the parity of real projective plane-curve intersections.

Equality with $ab$ need not hold. Choose $a=1$, $b=2$, with the line and conicOn the line, the conic equation becomes $x^2+y^2=0$, which has no nonzero real solution. Thus $\boxed{|A\cap B|=0\ne 2=ab}$. Both real curves are smooth and nonempty, and they are disjoint, so their real intersection is transverse vacuously. Even over the complex numbers they meet in two distinct transverse points $[1,i,0]$ and $[1,-i,0]$; the example is not a tangency or shared-component exception. On the unit sphere $F^2+g^2$ has a positive minimum, so disjointness also persists under sufficiently small real coefficient perturbations, so it is compatible with the generic real situation.

Use integral singular chains. The inclusion $Y\subseteq X$ identifies $C_k(Y)$ with the subgroup of $C_k(X)$ generated by the singular simplices whose images lie in $Y$. It is a chain subcomplex. DefineThis boundary is well defined because $\partial C_k(Y)\subseteq C_{k-1}(Y)$, and its square is zero because ${\partial }^2=0$. Its homology is, by definition, $H_k(X,Y)$. A relative cycle has a representative $c$ with $\partial c\in C_{k-1}(Y)$; representatives of the same relative homology class differ by $\partial b+y$, where $b\in C_{k+1}(X)$ and $y\in C_k(Y)$.

Inclusion and quotient induce homomorphisms $i_{\ast }:H_k(Y)\to H_k(X)$ and $j_{\ast }:H_k(X)\to H_k(X,Y)$. Define the relative homology connecting homomorphism byHere $\partial c$ is a cycle in $Y$ since ${\partial }^{2}c=0$. If $c$ is changed to $c+\partial b+y$, its boundary changes by $\partial y$, a boundary in $Y$, proving independence of representatives. The formula is additive, so $\delta$ is a homomorphism.

The long exact sequence in relative homology isand ends inWe prove all three types of exactness directly, so no unproved abstract lemma about chain complexes is required.

At $H_k(X)$, $j_{\ast }{i}_{\ast }=0$ because a chain in $Y$ is zero in the relative chain group. Conversely, let an absolute cycle $c$ have $j_{\ast }[c]=0$. The definition of a relative boundary gives $c=\partial b+y$ with $y\in C_k(Y)$. Since $c$ is a cycle, $\partial y=0$. Thus $[c]=i_{\ast }[y]$, proving $\ker j_{\ast }=\mathrm{im}{i}_{\ast }$.

At $H_k(X,Y)$, $\delta j_{\ast }=0$ because an absolute cycle has zero boundary. Conversely, suppose $\delta [c]=0$. Then $\partial c=\partial y$ for some $y\in C_k(Y)$. Therefore $c-y$ is an absolute cycle and $j_{\ast }[c-y]=[c]$. Hence $\ker \delta =\mathrm{im}{j}_{\ast }$.

At $H_{k-1}(Y)$, $i_{\ast }\delta =0$ because $\partial c$ is a boundary in $X$. Conversely, if a cycle $y\in C_{k-1}(Y)$ becomes zero in $H_{k-1}(X)$, then $y=\partial c$ for some $c\in C_k(X)$. That $c$ is a relative cycle, and $\delta [c]=[y]$. Thus $\ker i_{\ast }=\mathrm{im}\delta$.

These arguments also prove the assertions at shifted indices. Finally, every zero-dimensional relative cycle lifts to a chain in $C_0(X)$, and every zero-chain is an absolute cycle; this proves surjectivity onto $H_0(X,Y)$. All maps, representative changes, compositions and exactness assertions have consequently been justified from the chain definitions.

Work over $\mathbb{Q}$ and put $b_i=\dim H^i(M;\mathbb{Q})$. For a closed orientable six-dimensional manifold, Poincare duality gives $b_i=b_{6-i}$, soThe remaining Betti number is even. Indeed, the cup product pairingis nondegenerate by duality and skew-symmetric by graded commutativity, since $(-1{)}^{3\cdot 3}=-1$. If its dimension were odd, a matrix $A$ for it would satisfy $\det A=\det (-A^T)=-\det A$, forcing $\det A=0$ in characteristic zero, a contradiction. Thus $b_3$ is even andThis is the Euler characteristic parity in dimensions congruent to two modulo four. If the manifold has several components, apply the argument to each component and add.

The orientability assumption is essential. The Real projective space ${\mathbb{RP}}^6$ has one cell in each dimension from zero to six, givingIt is nonorientable: its double cover $S^6$ has the orientation-reversing antipodal deck transformation, of degree $(-1{)}^7=-1$.

For the construction, ${\mathbb{CP}}^3$ is a closed orientable real six-dimensional manifold with cells in dimensions $0,2,4,6$, hence Euler characteristic four. The product $S^3\times S^3$ is also closed and orientable, with Euler characteristic $\chi (S^3{)}^2=0$. The six-sphere has Euler characteristic two.

In dimension six, removing an open ball subtracts one from the Euler characteristic. To see this, write $M$ as the punctured manifold joined to a closed ball along $S^5$, whose Euler characteristic is zero, and apply the cellular inclusion-exclusion formula. Gluing two punctured manifolds along $S^5$ therefore givesA connected sum of $r$ copies of ${\mathbb{CP}}^3$ and $s$ copies of $S^3\times S^3$ haswith the empty sum interpreted as $S^6$. Given any even integer $n$, takeBoth are nonnegative integers and the displayed Euler characteristic equals $n$. Hence every even integer is the Euler characteristic of a closed oriented six-manifold, including zero and negative integers; the construction is connected.

Put $\Gamma ={\pi }_1(X,x)$ and $R=\mathbb{Z}[\Gamma ]$. Give $\mathbb{Z}$ its trivial left $R$-module structure, through the augmentation $\epsilon :R\to \mathbb{Z}$, $\sum n_{g}g\mapsto \sum n_g$. The universal cover $\tilde{X}$ has the lifted cell structure, with $\Gamma$ acting by deck transformations.

For each cell of $X$, choose one lifted cell and orient every translate compatibly with its projection. The translates form one copy of the regular $R$-module. Thusis a free left $R$-module, and its boundary is $R$-linear. The augmentation $C_0(\tilde{X})\to \mathbb{Z}$ sends every vertex to one. The contractibility of $\tilde{X}$ says that its reduced homology vanishes, so the augmented complexis exact. It is therefore a free resolution of the trivial module. By the definition of the Ext functor,

An $R$-linear cochain with values in the trivial module has the same value on every translate of a lifted cell. It is therefore exactly an ordinary integral cellular cochain on $X$. The correspondence does not depend on which lift was initially chosen. It also respects the coboundary: applying a constant-on-orbits cochain to the lifted cellular boundary sums precisely the incidence coefficients of the boundary in the base. Henceas cochain complexes, not just as graded groups. Cellular cohomology computes the cohomology of a cell complex, givingAt $i=0$ both sides are $\mathbb{Z}$, by connectedness and ${\mathrm{Hom}}_R(\mathbb{Z},\mathbb{Z})=\mathbb{Z}$. This is the cellular free resolution from a contractible universal cover description of group cohomology.

Use the curvature convention $R(X,Y)={\nabla }_X{\nabla }_Y-{\nabla }_Y{\nabla }_X-{\nabla }_{[X,Y]}$ of the earlier curvature calculation. A geodesic variation of $\gamma :[a,b]\to M$ is a smooth map $F:(-\epsilon ,\epsilon )\times [a,b]\to M$ with $F(0,t)=\gamma (t)$ such that every $t\mapsto F(s,t)$ is an affinely parametrized geodesic. Its variation vector field $J={\partial }_{s}F{|}_{s=0}$ satisfies the Jacobi equationIndeed $D_{s}T=D_{t}J$ by vanishing torsion tensor; differentiating $D_{t}T=0$ in $s$ and commuting the covariant derivatives gives this equation. A Jacobi field is a field satisfying this equation. Conversely, any prescribed pair $J(a),D_{t}J(a)$ is obtained by smoothly varying the initial point and velocity of a geodesic; smooth dependence of the geodesic equation then constructs a geodesic variation on the compact parameter interval, whose variation vector field is the prescribed Jacobi field by uniqueness of this linear equation.

For continuous piecewise smooth vector fields along a curve $V,W$ along $\gamma$, the Riemannian index form is the symmetric bilinear formContinuity at the breakpoints is part of the admissibility condition; without it an arbitrary jump could destroy the claimed index inequality. The curvature symmetries make $V\mapsto R(V,T)T$ self-adjoint. For fields with zero endpoints, this form is the second variation of geodesic energy $E=\frac{1}{2}{\int }_a^b|\dot{\gamma }{|}^{2}dt$. More explicitly, twice differentiating energy, commuting $D_s,D_t$ and integrating the term $⟨D_t(D_s{\partial }_{s}F),T⟩$ by parts leaves the displayed integral; $D_{t}T=0$ and fixed endpoints remove the remaining terms.

The endpoints $p=\gamma (a)$ and $q=\gamma (b)$ are conjugate points along $\gamma$ when there is a nonzero Jacobi field $J$ with $J(a)=J(b)=0$. Equivalently, the Riemannian exponential map at the initial velocity for this segment has singular differential: variations of initial velocity give precisely the Jacobi fields with zero initial value. For a unit-speed geodesic starting at $p$, its cut time isIf finite and the geodesic is defined there, its endpoint $\gamma (c(v))$ is the Riemannian cut point along that geodesic. Before the cut time the segment minimizes, at that time it still minimizes by continuity, and every positive extension ceases to minimize. This is the Riemannian notion, independent of whether deleting the endpoint disconnects the space.

Let ${\mathcal{H}}_0$ denote the continuous piecewise smooth vector fields along a curve with zero endpoints. The precise characterization at conjugate endpoints isThus Jacobi fields form the radical of the fixed-endpoint index form. For a smooth field $J$, integration by parts givesA Jacobi field makes both terms vanish. Conversely, if $I(J,W)=0$ for all tests, first take tests supported inside each smooth subinterval. The fundamental lemma gives the Jacobi equation on each piece. Integrating by parts piecewise then leaves, at an interior breakpoint $t_i$, the term $⟨D_{t}J(t_i^{-})-D_{t}J(t_i^{+}),W(t_i)⟩$. The values of the tests there are arbitrary, so all derivative jumps vanish. The continuous field and its continuous first covariant derivative solve the smooth Jacobi equation globally, making $J$ smooth by uniqueness of its initial-value problem. In particular, conjugacy means that this radical is nonzero. Merely imposing $I(J,J)=0$ is insufficient when the Riemannian index form is indefinite; a zero value of an indefinite quadratic form need not mean membership in its radical.

Now suppose there are no conjugate points on $(0,1]$. In a parallel orthonormal frame along $\gamma$, write the Jacobi equation as $j^{\prime \prime }+K(t)j=0$, where the symmetric matrix $K$ represents $Z\mapsto R(Z,T)T$. LetEvery Jacobi field vanishing at zero has components $j(t)=A(t)c$. The nonconjugacy assumption means $A(t)$ is invertible for every $0<t\le 1$, and hence the unique Jacobi field with the requested endpoint values isPut $U=W-V$, so $U(0)=U(1)=0$. Since $V$ solves the Jacobi equation, integration by parts gives $I(V,U)=0$ and thereforeTo establish the needed strict positivity, use the Riccati factorization of the Riemannian index form. Differentiation givesand the initial data make this conserved quantity zero. Consequently $S=A^{\prime }{A}^{-1}$ is symmetric and obeys the matrix Riccati equation $S^{\prime }+S^2+K=0$. For the components $u$ of $U$,Near zero, the differential equation gives $A(t)=t{I}_n+O(t^3)$ and $S(t)=t^{-1}{I}_n+O(t)$. Since $U$ is piecewise smooth and $U(0)=0$, $u(t)=O(t)$, so $u^{T}Su\to 0$. At time one the same boundary term vanishes because $u(1)=0$. Integration, including cancellation at all interior breakpoints by continuity, therefore givesIf equality holds, $u^{\prime }=Su$ on each piece, so $(A^{-1}u{)}^{\prime }=0$ on $(0,1]$. Continuity across the breakpoints gives $u=Ac$ with one constant vector $c$; $u(1)=0$ and invertibility of $A(1)$ force $c=0$. HenceThe nonzero endpoint condition is compatible with this proof, but the minimizing property itself holds for arbitrary specified final value.

For the relation to the Riemannian cut point, let $a>0$ be the first conjugate time along a unit-speed geodesic, and let $J$ be a nonzero Jacobi field on $[0,a]$ vanishing at both endpoints. Necessarily $D_{t}J(a)\ne 0$, otherwise uniqueness of the Jacobi equation would give $J=0$. Fix any later time $b>a$ for which the geodesic exists. Extend $J$ by zero to a continuous piecewise smooth field $Z$ on $[0,b]$. Then $I(Z,Z)=0$, while for any endpoint-vanishing smooth field $P$,Choose $P(a)=-D_{t}J(a)$, using a smooth bump times parallel fields, so this equals $-|D_{t}J(a){|}^2<0$. For small $\epsilon >0$,If a smooth variation is desired, smooth the single corner of this field while preserving the endpoints. The field and its derivative converge in the integral defining the Riemannian index form, so strict negativity persists. The second variation of geodesic energy then gives a fixed-endpoint path of smaller energy. By the Cauchy-Schwarz inequality, $L^2\le 2bE$ for paths on $[0,b]$, with equality for the original unit-speed geodesic; the new path is therefore shorter. This proves loss of geodesic minimality beyond a conjugate point, and henceA missing conjugate time is interpreted as infinity. If a finite conjugate time exists and the geodesic continues past it, every later segment fails to minimize, so its first loss of minimality occurs no later than that conjugate time.

Finally, the cut-point dichotomy for complete Riemannian manifolds is the following characterization, stated without proof. On a complete connected Riemannian manifold, the cut time is the first time at which either the endpoint is conjugate to $p$ along the given geodesic, or a distinct minimizing geodesic from $p$ reaches the same endpoint with the same length. Thus a finite Riemannian cut point is either the first conjugate point or the meeting point of at least two distinct minimizing geodesics from the initial point; both alternatives may occur together. Completeness is the hypothesis guaranteeing that competing minimizing geodesics exist at a nonconjugate cut endpoint. It is not needed for the preceding local index-form argument on an existing segment.

Work on a connected, finite-dimensional Riemannian manifold without boundary. It is geodesically complete if every maximal affinely parametrized geodesic has parameter domain $\mathbb{R}$. Equivalently, for every $p$ the Riemannian exponential map ${\exp }_p$ is defined on all of $T_{p}M$: any finite affine time can be rescaled to time one, and negative times correspond to reversing the initial velocity.

The Hopf-Rinow lemma in its minimizing-geodesic form says: if ${\exp }_p$ is defined on all of $T_{p}M$ for one point $p$, then every $q$ can be joined to $p$ by a geodesic of length $d(p,q)$. We first establish the local ingredient, the distance splitting through a small geodesic sphere. If $D=d(p,q)>0$, choose $0<\epsilon <D$ sufficiently small that the closed normal ball about $p$ is compact and radial lengths give Riemannian distance, as proved using the Gauss lemma. Take paths from $p$ to $q$ of lengths tending to $D$. Their first intersections $x_j$ with its boundary sphere have a subsequence converging to some $x$ on the sphere. Their lengths are at least $\epsilon +d(x_j,q)$, giving $D\ge \epsilon +d(x,q)$. The triangle inequality gives the reverse inequality. ThereforeThis splitting uses only a compact local geodesic sphere, not completeness or the existence of a globally minimizing path.

We also need that a minimizing broken geodesic has no corner. To see this directly, let the incoming and outgoing unit tangents at a junction be $u$ and $v$. Choose one nearby point on each segment so that they and the junction lie in a common convex normal neighbourhood. Move the junction with velocity $z$, joining it to those two fixed points by the unique short geodesics. The Gauss lemma, or the first variation formula, gives the derivative of the sum of their lengths as $⟨u-v,z⟩$. If $u\ne v$, setting $z=v-u$ makes this derivative $-|u-v{|}^2<0$, contradicting minimality. Thus the tangent vectors agree, and uniqueness for the geodesic equation makes the broken path one smooth geodesic.

Now choose the unit-speed radial geodesic $\gamma$ through the point $x$ obtained by the distance splitting, with $\gamma (0)=p$ and $\gamma (\epsilon )=x$. By the hypothesis on ${\exp }_p$, it is defined at least up to time $D$. DefineThis is a nonempty closed set. If $t\in A$, the triangle inequality and the length of $\gamma {|}_{[0,t]}$ giveso $d(p,\gamma (t))=t$: that geodesic segment minimizes. Let $t_{\ast }=\max A$. If $t_{\ast }<D$, apply the distance splitting through a small geodesic sphere at $\gamma (t_{\ast })$ with radius $0<\delta <D-t_{\ast }$. There is a point $y$, joined to $\gamma (t_{\ast })$ by a short minimizing radial geodesic $\sigma$, for whichThe broken path $\gamma {|}_{[0,t_{\ast }]}\ast \sigma$ has length $t_{\ast }+\delta$. Moreover,Consequently that broken path minimizes. It has no corner, so $\sigma$ is the continuation of $\gamma$ and $y=\gamma (t_{\ast }+\delta )$. The last distance equality puts $t_{\ast }+\delta$ in $A$, a contradiction. Hence $D\in A$ and $\gamma (D)=q$. This proves the Hopf-Rinow lemma, with the case $p=q$ supplied by the constant geodesic.

The Hopf-Rinow theorem asserts the equivalence of

These conditions imply the existence of a minimizing geodesic between any two points. Here is the deduction, including the implication back to geodesic completeness. If the manifold is geodesically complete, the Hopf-Rinow lemma applies at each point. For every $R\ge 0$,One inclusion follows from the length of the radial geodesic; the other follows by taking an initial velocity of a minimizing one. The tangent-space ball is compact, so the metric ball is compact as its continuous image. Every closed bounded set is a closed subset of such a ball and is compact. A Cauchy sequence is bounded and therefore has a convergent subsequence in a compact ball; the Cauchy property makes the whole sequence converge. This proves completeness of the Riemannian distance.

Conversely, assume metric completeness and let $\gamma :[0,b)\to M$ be a geodesic with $b<\infty$. Its speed $c$ is constant, so $d(\gamma (s),\gamma (t))\le c|s-t|$. Completeness therefore gives a limit $q$ as $t\uparrow b$. Eventually $\gamma$ lies in a coordinate neighbourhood with compact closure on which the Riemannian metric is uniformly comparable to the Euclidean metric. Its coordinate velocity is bounded by the constant-speed estimate. The Christoffel symbols are bounded there, so the geodesic equationmakes its coordinate acceleration bounded. Thus the coordinate velocity also has a limit $v$ at time $b$. The local existence and uniqueness theorem for smooth ordinary differential equations extends the solution from initial data $(q,v)$ past $b$, contradicting maximality. Reversing time treats a finite left endpoint. Hence metric completeness implies geodesic completeness, and the preceding argument supplies compactness of closed bounded sets. Compactness of closed bounded sets already implies metric completeness by the Cauchy-sequence argument, completing all equivalences.

In fact, the one-point hypothesis of the Hopf-Rinow lemma suffices for the theorem: the same closed-ball image argument at that point puts every Cauchy sequence in a compact ball, giving metric completeness and then geodesic completeness at every point. By contrast, the existence of a minimizing geodesic for every pair alone does not imply completeness: an open Euclidean ball has minimizing straight segments between all its points but has Cauchy sequences converging to its missing boundary.

The Riemannian exponential map at $p$ is ${\exp }_p(v)={\gamma }_v(1)$, where ${\gamma }_v$ is the affinely parametrized geodesic with ${\gamma }_v(0)=p$ and ${\dot{\gamma }}_v(0)=v$. Its derivative at zero is the identity. The inverse function theorem therefore gives a normal neighbourhood $U={\exp }_p(V)$, where $V\subset T_{p}M$ is an open star-shaped neighbourhood of zero on which ${\exp }_p$ is a diffeomorphism. Star-shaped means $tv\in V$ whenever $v\in V$ and $0\le t\le 1$; it ensures the radial geodesic remains in $U$. This is distinct from a convex normal neighbourhood, which imposes a unique short geodesic between every pair of its points.

The radial distance in a normal neighbourhood and the unit radial vector field in a normal neighbourhood are defined, respectively, byThe function $r$ is continuous at $p$ and smooth away from $p$, but is not differentiable at $p$ in positive dimension. Some conventions use the scaled radial field $W_q=(d{\exp }_p{)}_{v}v$ and the smooth radial function $r^2/2$ instead; the associated gradient identities will distinguish them.

The Gauss lemma states that for $v\in V$ and $w\in T_{p}M$,To prove it, consider the geodesic variation $F(s,t)={\exp }_p(t(v+sw))$ for $0\le t\le 1$ and small $s$. Set $T={\partial }_{t}F$ and $J={\partial }_{s}F$. The Levi-Civita connection has zero torsion tensor, so $D_{t}J=D_{s}T$. Each $t$-curve is a geodesic, hence $D_{t}T=0$, and its squared speed is the constant $|v+sw{|}^2$. At $s=0$,Since $J(0)=0$, integration gives $g(T,J)=t{g}_p(v,w)$. At $t=1$, $T=(d{\exp }_p{)}_{v}v$ and $J=(d{\exp }_p{)}_{v}w$, proving the Gauss lemma. Taking $w=v$ shows $|E|=1$; taking $w\perp v$ shows that $E$ is orthogonal to the images of tangent vectors to the tangent-space sphere, hence to the geodesic spheres in $U$.

For $u\in T_{p}M$ of unit length, write the unit-speed radial geodesic as ${\gamma }_u(t)={\exp }_p(tu)$. Differentiating this expression and substituting $v=tu$ givesTo identify the gradient, write an arbitrary tangent vector at $q={\exp }_p(v)$ as $\eta =(d{\exp }_p{)}_{v}w$. Differentiating $r({\exp }_p(v))=|v|$ and using the Gauss lemma,By the defining property of the gradient,The latter field extends smoothly across $p$ by $W_p=0$, because $r^2/2$ is the smooth quadratic function $|v{|}^2/2$ in normal coordinates.

The same calculation also proves local radial minimization: for a piecewise smooth curve in the normal neighbourhood, $|dr(\dot{\alpha })|\le |\dot{\alpha }|$, so its length from $p$ to $q$ is at least $r(q)$, attained by the radial geodesic. Take a small tangent ball with closure inside a larger normal ball. Any path leaving the small ball before reaching a point inside it has already accumulated length at least its radius. Therefore, for points in the small ball, $r(q)=d(p,q)$ even when competitors may leave the neighbourhood. This local argument does not assert that $r$ equals global Riemannian distance throughout every arbitrarily large star-shaped normal neighbourhood.